TOKYO (Reuters) - LVMH, the world's biggest luxury goods group, has scrapped a planned Louis Vuitton flagship store in Tokyo's glitzy Ginza shopping district, as a spreading recession takes the luster off designer brands.
LVMH has broken off talks over opening a store in a planned building in Ginza due for completion in 2010, an official for the developer, Hulic Co Ltd, said on Tuesday.
An LVMH spokesman said it had withdrawn from the project but declined to explain the decision or comment on its plans.
"We have not been able to reach an agreement with Hulic on a store development project in Ginza," said Yuri Matsueda of Burson-Marsteller, a public relations agent for Louis Vuitton Japan.
Awash with cash from a global luxury boom, fashion and jewelry brands poured money into lavish retail spaces in upmarket Tokyo neighborhoods such as Ginza and Omotesando over the past two years.
Earlier this year, media reports had talked of a 10-floor luxury Louis Vuitton emporium and restaurant to rival opulent new Armani and Bulgari towers in Ginza.
A weak yen also encouraged such investments, tempting Japanese fashionistas to shop at home rather than Europe, luring buyers from all over Asia to Tokyo and offering foreign brands bargains in the property sector.
While luxury goods firms gave optimistic forecasts for their trade as recently as May, banking on the super-rich to keep spending despite the economic slowdown, the mood has turned as the impact from the global financial crisis has grown.
Japan, the United States and much of Western Europe is in recession and growth is slowing in emerging markets such as China.
Several companies have cut their outlooks and expect a grim Christmas season followed by an even gloomier 2009.
LVMH had appeared to be weathering the downturn better than others. It said in October that sales had risen sharply in the first week of that month, and sales in emerging markets were holding up.
The Nikkei business daily reported on Tuesday Louis Vuitton Japan had planned to open what would be one of the world's biggest stores in Ginza but had to give up due to severe slump in luxury spending.


text:
Sir Richard Baker  (by 1530 – 27 May 1594), was an English politician.

Family
Richard Baker was the eldest son of Sir John Baker, Chancellor of the Exchequer and his second wife Elizabeth Dineley. He had a younger brother, John, and three sisters, one of whom, Cecily, was the wife of Thomas Sackville, Lord Buckhurst.

Career
Baker studied law at the Inner Temple. He succeeded his father in 1558, and was knighted on 31 August 1573.

He was elected a Member (MP) of the Parliament of England for Horsham in April 1554, Lancaster in November 1554, New Romney in 1555 and New Shoreham in 1558. He was a Justice of the Peace for Kent from 1558 and appointed High Sheriff of Kent for 1562–63 and 1582–83.

Marriages and issue
Baker married firstly Katherine Tyrrell, the stepdaughter of Sir William Petre. She was the daughter and heiress of John Tyrrell (d.1540), esquire, of Heron in East Horndon, Essex, (eldest son of Sir Thomas Tyrrell and Constance Blount), and his wife, Anne Browne, the daughter of Sir William Browne, Lord Mayor of London, by his second wife, Alice Keble (d. 8 June 1521), the daughter of Henry Keble (1452 - April 1517), Lord Mayor of London, by whom he had two sons and a daughter.

Baker married secondly Mary, the daughter of John Gifford of Weston Subedge, Gloucestershire, with whom he had another 2 daughters. He was succeeded by his son Thomas.

One of his daughters by Mary was Chrysogna Baker (d. 1616) who married Sir Henry Lennard, 12th Baron Dacre and had issue.

Footnotes

References

External links
Will of Sir Richard Baker of Cranbrook, Kent, proved 13 June 1594, PROB 11/84/37, National Archives Retrieved 10 July 2013

1594 deaths
Members of the Inner Temple
High Sheriffs of Kent
English MPs 1554
English MPs 1554–1555
English MPs 1555
English MPs 1558
Year of birth uncertain

redpajama_set_name: RedPajamaWikipedia


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text:
Home » Love Lingers Here: Intimate Enduring Relationships–Essay IV: The New Self and Founding Story
Child / AdolescentSystem DynamicsPersonalityCouples & Family Psychology
Love Lingers Here: Intimate Enduring Relationships–Essay IV: The New Self and Founding Story
Burt and Jill, for instance, talked about their decision to have a child. The unified story focused on a particularly special evening at a bed-and-breakfast inn where they made wonderful, spontaneous love. Their unified story tells of this special, romantic evening as the time when their first child was conceived. Both partners agreed on this segment of their unified story. Yet, when Burt talked about this event, he focused on the events that led up to the weekend. He described a particularly painful argument that they had one week prior to their romantic evening, concerning their finances and his current job. Burt wasn't certain that they could afford to have a child, while his wife, Jill, was convinced that they could afford a child, if he would "get off his rear end" and find a higher paid job. The romantic evening was special for Burt because she began the evening by apologizing for pushing him too hard, while he made a commitment to her to begin a job-search. And that evening, their daughter, Allyson, was conceived—at least according to their unified story.
Conversely, Jill spoke of a conversation she had with her sister the day before her romantic evening with Burt. She described her sister's painful revelation that she had just found out that she and her husband could never give birth to their own children, and that they would have to look to adoption if they wanted to raise children. At this moment, according to Jill, she realized how fortunate she and Burt were. She decided then and there that she should become pregnant and that she needed to be supportive of Burt, so that he wouldn't feel under as much .financial pressure and would agree to have a child. Jill speaks of her commitment to working harder in her own job, in order to get a raise, and to scrimp and save at home so that she and Burt could have a child. For Jill, this was the critical moment, leading up to her romantic evening with Burt, and her pregnancy.
The story that Jill and Burt told us about the conception of their daughter was very important to them and revealed much about the character of their relationship. Yet, this was not the only story they told us. They told us many other stories during their interview, as did most of the couples we interviewed. In fact, for Jill and Burt — as was the case for most of the couples we interviewed — the most revealing story concerned not the conception of their daughter, but rather the beginning of their own relationship. Virtually all of the couples we interviewed had a wonderful story about their meeting and their formation as a couple. This founding story often established the basic norms and values of their relationship, and continues to have a strong influence on the way in which they relate to one another.
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William BergquistWilliam Bergquist, Ph.D. An international coach and consultant in the fields of psychology, management and public administration, author of more than 50 books, and president of a psychology institute. Dr. Bergquist consults on and writes about personal, group, organizational and societal transitions and transformations. His published work ranges from the personal transitions of men and women in their 50s and the struggles of men and women in recovering from strokes to the experiences of freedom among the men and women of Eastern Europe following the collapse of the Soviet Union. In recent years, Bergquist has focused on the processes of organizational coaching. He is coauthor with Agnes Mura of coachbook, co-founder of the International Journal of Coaching in Organizations and co-founder of the International Consortium for Coaching in Organizations.
View all posts by William Bergquist

redpajama_set_name: RedPajamaCommonCrawl


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Take the challenge to do body squats Monday through Friday the enitre month of May. Prizes will be given to all who complete the challenge. Starting on May 1, do 20 squats and then increase by 10 reps every day. If you miss a day you can make it up the following day. All squats must be observed by an MWR staff member. Register by April 20. Call 757-688-7451.


Today in Talking Money we'll be looking at Jeff Bezos, founder of Amazon.
Today in Talking Travel we'll be looking at travel safety for this summer period.
Today in Talking Health we'll be looking at homeopathy.
Today in Talking Money we'll be looking at pensions.
Today in Talking Travel we'll be visiting the most photogenic countries for national photo week.
Today in Talking Life we'll be looking at efficient packing for the holidays.
Today in Talking Tech we'll be looking at home robots.
Today in Talking Health we'll be looking at body shaming.
Today in Talking Money we'll be looking at zero hour contracts.
Today in Talking Travel, we'll be hitching a ride through the Australian Outback.
Today in Talking Life, we'll be looking at the obsession of instagram.
Today in Talking Tech, we'll be looking at Affiliate Marketing.
Today in Talking Health, we'll be looking at female genital mutilation.
Today in Talking Money, we'll be looking at the latest Brexit developments.
Today in Talking Travel, we'll be going to Saint Martin in the Caribbean.
Today in Talking Life, we'll be looking at community spirit in the UK.
Today in Talking Tech, we'll be looking at public Wi-Fi and the High Street.
Today in Talking Health, we'll be looking at In Vitro Fertilization.


text:
package org.apache.cordova.wifiinfo;

import org.apache.cordova.CallbackContext;
import org.apache.cordova.CordovaPlugin;
import org.apache.cordova.PluginResult;
import org.json.JSONArray;
import org.json.JSONObject;
import org.json.JSONException;

import android.content.Context;
import android.net.wifi.ScanResult;
import android.net.wifi.WifiInfo;
import android.net.wifi.WifiManager;
import java.math.BigInteger;
import java.net.InetAddress;
import java.util.Arrays;
import java.lang.StringBuilder;

public class cordovaWifiInfo extends CordovaPlugin {

	public cordovaWifiInfo() {
	}

	@Override
	public boolean execute(String action, JSONArray args, CallbackContext callbackContext) throws JSONException {
		if (action.equals("getWifiInfo")) {
			Context context = cordova.getActivity().getApplicationContext();
			PluginResult pluginResult = new PluginResult(PluginResult.Status.OK, this.loadData(context));
			pluginResult.setKeepCallback(true);
			callbackContext.sendPluginResult(pluginResult);
			return true;
		}
		return false;
	}

	private String parseIP (int intIP) {
		try {
			byte[] bytes = BigInteger.valueOf(intIP).toByteArray();
			int i = 0;
			int j = bytes.length - 1;
			byte tmp;
			while (j > i) {
				tmp = bytes[j];
				bytes[j] = bytes[i];
				bytes[i] = tmp;
				j--;
				i++;
			}

			return InetAddress.getByAddress(bytes).toString().replace("/", "");
		} catch (Exception e) {
			return "";
		}
	}

	private JSONObject loadData(Context context) {
		WifiManager wifiManager = (WifiManager) context.getSystemService(Context.WIFI_SERVICE);
		WifiInfo wifiInfo = wifiManager.getConnectionInfo();
		JSONObject obj = new JSONObject();
		try {
			JSONObject lan = new JSONObject();
			lan.put("BSSID", wifiInfo.getBSSID());
			lan.put("HiddenSSID", wifiInfo.getHiddenSSID());
			lan.put("SSID", wifiInfo.getSSID().replace("\"", ""));
			lan.put("MacAddress", wifiInfo.getMacAddress());
			lan.put("IpAddressInt", wifiInfo.getIpAddress());
			lan.put("IpAddress", parseIP(wifiInfo.getIpAddress()));
			lan.put("NetworkId", wifiInfo.getNetworkId());
			lan.put("RSSI", wifiInfo.getRssi());
			lan.put("LinkSpeed", wifiInfo.getLinkSpeed());
			obj.put("lan", lan);

			if(wifiManager.getScanResults() != null){
				JSONArray networks = new JSONArray();
				for (ScanResult scanResult : wifiManager.getScanResults()) {
					JSONObject ap = new JSONObject();
					ap.put("BSSID", scanResult.BSSID);
					ap.put("SSID", scanResult.SSID);
					ap.put("frequency", scanResult.frequency);
					ap.put("level", scanResult.level);
					//netwrok.put("timestamp", String.valueOf(scanResult.timestamp));
					ap.put("capabilities", scanResult.capabilities);
					networks.put(ap);
				}
				obj.put("networks", networks);
			}
		} catch (Exception e) {

		}
		return obj;
	}
}

redpajama_set_name: RedPajamaGithub


text:
Blue Line Poll: Smart Regs
By Alan Boles
Landlords clashed over the likely effects of the City of Boulder's proposed SmartRegs (which will be debated by the City Council at a second reading on July 6) at a PLAN-Boulder County forum on the subject on June 21. Francoise Poinsatte, a forum panelist, rental property owner and former member of the Boulder City Council, claimed that compliance with the proposed regulations would be relatively easy and inexpensive for landlords and enhance their property values. In contrast, Sheila Horton, an audience member who is the executive director of the Boulder Area Rental Housing Association, argued that the regulations would create major financial burdens and no benefits for landlords.
Francoise declared that she has always taken pride in the physical condition of her rental properties. She said that she has already made significant improvements to them, including energy efficiency improvements, and welcomes the prospect of improving them still more. She noted that the proposed regulations emphasized relatively inexpensive measures–such as insulating walls, attics and under-floor crawlspaces, and stopping air leakage. She recounted that she had enlisted her 19-year old son to install insulation in some of her rental homes and that the insulation material itself had been fairly cheap to buy. She also observed that a high proportion of Boulder's rental housing stock is constructed in the same way and that consequently energy improvements could become relatively standardized, thereby perhaps reducing costs. Francoise asserted that the re-sale value of her properties will rise as they use less energy.
Sheila, however, implied that the re-sale prices of rental properties are determined only by the gross amount of rents they can command in the market and contended that rents have been under severe pressure in the City of Boulder recently. She said that 2,500 units of rental housing had been built in the City during the past two years. That increase in supply has depressed rental rates. She claimed that landlords could not pass on the costs of compliance with SmartRegs to their tenants, even when the tenants' utility bills fall, because market competition with their peers would not allow them to do so. Sheila also asserted that landlords around the nation are reeling from the expense of complying with new regulations on lead-based paint, which the Environmental Protection Agency put into effect in April, 2010, and apply to buildings constructed before 1979.
Another panelist at the forum, Yael Gichon, who is a member of the City of Boulder's Environmental Affairs Office, said that the proposed SmartRegs have been designed to promote the energy efficiency measures that produce the most benefit for the least cost. She said that they had been devised after a City consultant had conducted case studies of seven different types of rental housing. The proposed regulations would require each rental unit to obtain a "score" of 100 over two rental licensing cycles and at least 50 during each cycle. Each rental licensing cycle last four years. Yael asserted that every rental unit in the case studies already qualified for a "score" of at least 50. She also stated that the regulations may provide for a yearly spending limit on energy improvements by landlords and that they will allow exemptions for two types of "hardship": financial and technical.
A third panelist, Tim Hillman, a tenant, claimed that a "market failure" has occurred in rental housing, because the tenants typically pay the utility bills, while only the landlords have the legal authority and the incentive to spend money on the housing to improve its energy efficiency. Consequently, regulation is needed to correct the market failure. He also observed that over 50 percent of the City's dwelling units are rental, which percentage makes the energy impact of the market failure significant. He predicted that the proposed SmartRegs, if they are passed, will stimulate the local economy.
Tim cautioned that the behavior of tenants, as well as landlords, needs to change to reduce the waste of energy. He said that tenant education would be helpful, and he commended the University of Colorado for energy educational programs it has conducted for students. However, he recognized that education alone requires time to change behavior. He enthusiastically endorsed close cooperation between landlords and tenants to conserve energy.
The fourth panelist, Ann Livingston, who is Boulder County's Sustainability Coordinator, said that rental properties in the City of Boulder qualify for Boulder County's Climate Smart loan program, as well as for Xcel's energy rebates. Neither credit scores nor loan-to-value ratios are considered by the County in issuing a Climate Smart loan. The minimum size of a Climate Smart loan is $3,000 and the maximum is $50,000 or 20 percent of the assessed value of the property, whichever is lower. The loans may only be used to pay for fixtures, not appliances. They bear the same interest as the bonds which Boulder County sold to finance them, plus a small administrative fee; and they are secured by a lien on the improved property, which becomes due at the time of sale.
Ann reported that Boulder County has issued $10 million in Climate Smart loans during the past 12 months. The average cost of energy improvements per dwelling unit has been $4-5,000.She cautioned that both Fannie Mae and Freddie Mac, the gigantic, Federally sponsored mortgage insurers, have recently raised objections to the way that Climate Smart loans are secured.
She also noted that the County's PACE energy program provides "micro-loans" from $500 to $3,000 for terms of up to three years at low or no interest. Ann disclosed that Boulder County is currently considering applying for a $20 million block grant, $8 million of which would be reserved as a pool to help finance residential energy improvements. According to Ann, the County is planning to participate with the City in the much-anticipated "Two Techs and a Truck" program. That program is expected to be inaugurated in the next few months and will be available for rental, as well as owner-occupied, properties.
What do you think? Should landlords be required to make energy efficiency upgrades to their properties? Vote in the Blue Line Poll today!
New Era News | Renters With Benefit$
Colorado Independent | New Era knocks on doors for SmartRegs in Boulder
Dithering Against Climate Change
Sustainable City Network | Portland Energy Retrofit Program Goes Statewide
Write City Council to Support SmartRegs!
Author(s): Alan Boles
Found in section: Featured, Poll
Tags: climate smart loan, energy, rental, smart regs
Previous Topic: City Council Study Session
Next Topic: PLAN-Boulder Forum: Conoco Phillips
3 Responses to "Blue Line Poll: Smart Regs"
Mark Gelband says:
Now that climate smart loans aren't an option, isn't it time for the Anne Livingston and the city to update their data? And with the winds of energy change blowing Xcel and our franchise agreement out the door, who will be "paying" for those rebates?
Even with a $25,000 rebate and a $9,000 dollar federal tax credit, the payback on my recently installed PV system will be about 6-9 years. For three months I have produced more electricity than I use, but who in our community can afford a system without those credits and rebates? Even with them it's a big outlay and small return on investment. That doesn't even consider that the panels are manufactured in China, and the environmental impact of productiona and import?
Which brings me to Ms. Poinsatte, whose article about upgrading her rentals I have read. If the insulation she used from Home Depot were fiberglass batting, which seems likely since she is claiming it is relatively inexpensive, her choice in insulation and store are both questionable choices for one who is concerned with the environment and local economy, but not so for one who cares about her personal wealth. More environmental choices in insulation – soy-based spray foam or recycled cellulose, paper or cotton, are considerably more expensive.
And Yael asserted that all the units studied scored 50 already, did she mention that two of the seven they retrofit never made it to 100 points? When Mr. Hillman talks of behavior, are we going to make it illegal for folks to open their windows in the winter with the heat on?
Ultimately, conservation is a matter of personal choice. And even the proponents of more mommy government regulation could turn off their sprinklers, live in more modest homes, recycle more, consume less, hang their clothes to dry, etc. Do we really want to live in a community that dictates what others do without first looking in the mirror?
Tweets that mention Blue Line Poll: Smart Regs | The Blue Line -- Topsy.com says:
[…] This post was mentioned on Twitter by Mathew Kenney. Mathew Kenney said: Blue Line gives the skinny on the SmartRegs forum. Check it out! http://fb.me/w45yIQDm […]
Zane Selvans says:
This is a classic example of split incentives. Landlords will never make improvements (such as energy efficiency measures) which only benefit tenants and not themselves of their own accord. Applied uniformly, these higher standards will result in higher rents, but lower utility bills for renters (and less energy used overall), which is exactly what we should be aiming for. Hopefully we can find a financing mechanism which will work, even with the mess that the feds have foisted on us through Fannie and Freddie and the ClimateSmart loans.
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The School of Art and Design of Sant Cugat (EAD Sant Cugat) is a public school closely linked to its city, with a vocation for public service that promotes specialized training in official studies of Plastic Arts and Design in the field of Graphic and Audiovisual Communication. We offer a Vocational Education and Training course (CFGM in Catalan and Spanish) with specialization in Support to the Printed Graph Product, and Higher Education courses (CFGS in Catalan and Spanish) specialized in Animation, Illustration, Printed Graphics and Interactive Graphics.
The pedagogical and organizational autonomy of the EAD Sant Cugat and the teamwork of the teaching staff allow for the development of flexible measures that can meet the demands of the students.
Our studies facilitate continuous training with an international projection, connecting the Vocational Education and Training course to the Higher Education courses, and thus offering the possibility of obtaining two degrees in three years while supporting international mobility with the Erasmus Plus program.
The DNA of the school is formed by its dynamic, close and inclusive character and the Educational Project, which is based on quality and transversal education, and which aims to train professionals who are creative, critical and committed to the present surrounding them. The school is next to Barcelona, located in a unique space: on the site of a modernist tower called Casa Monaco. The EAD is in a quiet and green environment, close to the Collserola Natural Park, and five minutes from the center of Sant Cugat. It is well communicated for both public and road transport and it is in a dynamic university environment that offers all the services students may need.
We consider that the students who stand to gain the most from this excellent opportunity are those in their second year and third year of their studies.
The EAD Sant Cugat offers different itineraries among the Higher Education courses, and all of them are authorized by the Department of Education. This allows students to obtain two qualifications in three academic years (2x3) with a total of 180 ECTS credits. In the third year students can take their last semester abroad with an Erasmus program for either studies or internships.
The aim of the Erasmus program is to improve the students' performance, their career prospects, and to enhance their motivation and initiative. It is also to increase autonomy and self-esteem, to improve their foreign language skills, and to achieve greater intercultural awareness.
For the EAD Sant Cugat, the Erasmus project is a gateway to the improvement of educational offer, the birth of new projects and cultural exchange.
Els estudiants que gaudeixin d'una beca Erasmus+ SMS/SMT rebran un curs d'idioma on-line a través de la plataforma OLS (Online Linguistic Support) - que els ajudarà a preparar-se per la seva estada al país estranger.
Learning Agreement Proposal. We recommend choosing subjects from the seconds years. See curricula below (Our Studies Tab).
Documentation to support adequate proficiency in the language of instruction (Spanish or Catalan).
Applications should be sent via e-mail: mobility@escoladart.com. Only submissions in digital format will be accepted (pdf format and compressed into a zip clearly labelled with the name of the student).
Only the signed application form will be sent per post mail to EAD Sant Cugat.
Spring Semester: from February to June.
In the EAD Sant Cugat lessons are taught in Catalan or in Spanish. We recommend that incoming students have some knowledge of Spanish or Catalan. The desirable level is B1.
To find out more about the application process, contact the Erasmus institutional coordinator at your home institution.
Our specific and vocational courses aim to train the students to carry out a professional activity in the field of Graphic and Audiovisual Communication, either through narration, design or the graphic and audiovisual processing of information. Fixed, interactive or moving images are used, with a final printed and/or audiovisual result. One of the pedagogical aims of the school is to train students as creative professionals capable of self-management and of delving into their own knowledge and skills, so they can adapt the training acquired to their interests and respond to the changing conditions of the current labor market.
For this reason, the EAD offers different itineraries, all authorized by the Department of Education —among the Higher Education Courses— which allow students to obtain two certificates in three academic years (2×3) by extending the curriculum with the modules of the center and the tutorial monitoring of the Integrated Project. In order to make this curricular design possible, the EAD starts the Higher Education courses with a first introductory year to the family of Graphic and Audiovisual Communication. The second year is specialized and it also includes the Integrated Project and the Practical Working Training (internship) in work centers. The course finishes with a third year of another specialization where students can take their last semester abroad with an Erasmus program for either studies or internships.
Click here to download the EAD's Higher Education Courses PDF.
Click on the image to download the EAD's Higher Education Course PDF.
Create, design, organise, manage and supervise animation projects. Model and represent characters, objects, scenarios and backgrounds (2D and 3D). Animate, illuminate and place virtual cameras. Analyse proposals and carry out documentary and graphic research in different fields.
Design, organise, manage, supervise and analyse interactive graphic projects applied to different fields. Web design and development, production of interactive content: accessibility, usability and interfaces design.
Design, organise, develop, and supervise the production of printed graphic projects. Practice creative and artistic direction of brands, products, services and companies. Analyse, create and propose graphic products applied to different fields. Choose the use of graphic, typographic or photographic resources for different applications and prepare the final artwork for printing.
Realize original illustrations for a variety of purposes: science, technology, architecture, interior design, fashion, publicity, product, editorial, and press. Master traditional and digital techniques and apply them to interactive, printed and/or foldable media.
(A32, A34, A22) equipped with drawing tables, individual lighting, wifi and power access points for laptops and general computer, audio and projection system for theoretical and practical classes. These classrooms are suitable for artistic and technical drawing (with floor for life model), techniques of graphic expression, calligraphy, etc. Capacity: 30 students.
(A21) has different types of computers and scanners for the preparation of originals for printing. Used by Graphic Print modules. Capacity: 15 students.
(A1-A13) general computer equipped with audio system and projection. Capacity: 30 students.
(A31) equipped with complete lighting system, backdrops, filters, diffusers, stands, with store of equipment and accessories.
(A4) individual editing suite with two high-end computers, dedicated to video editing and 3D (Island edition) and studio filming and editing, stop-motion (3 individual sets), work tables and tools.
(A19) space attached to the building with outdoor terrace. It has screw press, exposure unit, and screens, allowing for traditional transfer techniques: intaglio and relief printing, screen printing and photo-etching.
(A0) common areas such as access hall, stairs and corridors are used as spaces for exhibition of student work.
* All classrooms have access to the internal computer network (intranet) and high-speed internet through a wireless system and/or high-performance cabling and capacity.


Breast cancer is a cancer that begins in the cells of the breast. The malignant cancerous breast cells and healthy tissues can be spread around. This is referred to as metastasis.
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Palliative care Which Doctor To go to breast cancer?
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text:
Alder Hey Children's Hospital
Liverpool celebrities give star backing to appeal in memory of tragic teenager Rose Farley
LIVERPOOL celebrities have thrown their backing behind a fundraising campaign in memory of tragic teenager Rose Farley.
John Siddle
Rose Farley who died hours after a party in Vauxhall
Sign up to FREE daily email alerts from Liverpool Echo - Liverpool FC
The budding actress, 15, died after falling ill at a birthday party in June.
Mystery still surrounds the death of the popular West Derby girl, whose drink may have been spiked.
Her friends and family have now launched the Rose Farley Appeal to support Alder Hey hospital's Imagine Appeal.
Paul Broughton, best known as Eddie Banks in Brookside, and Good Cop actor Mark Womack are both close friends of Rose's dad Chris.
They both helped secure a number of stellar auction items which will go under the hammer to raise cash for the appeal this Sunday.
Among the exciting prizes is the chance to bid for tickets to meet comedian John Bishop at one of his Liverpool shows, score a walk-on part and credit in the fourth series of city producer Colin McKeown's TV drama Moving On and bag a Liverpool FC shirt signed by Steven Gerrard.
There is also the chance to meet the stars of West End musical Wicked.
Paul told the ECHO: "We started this off as a charity football match and it has just spiralled. It has gone ballistic.
"People went out of their way to help. The generosity and spirit in this city never ceases to amaze me.
"Rose's death has struck a chord with people and hopefully we will raise a lot of money in her name."
Mark, who starred with John in the Ken Loach film Route Irish last year, said: "I have known Chris for a long time and I was so shocked when I heard the news about what happened. I wanted to do anything I could.
"John Bishop has been absolutely brilliant and donated four tickets and a meet-and-greet at his ECHO Arena shows in October, and Steven Gerrard has been brilliant too.
"He donated some signed shirts – in fairness, both of the Liverpool clubs have really made an effort. Rose's death was a tragedy but people are really responding to what we are doing and I am sure that will continue."
The fundraiser starts on Sunday at 12pm, where amateur football teams Rob Roy FC and New Campfield will face off at Hope university fields in Childwall.
The auction will take place afterwards at Chillies wine bar in High Street, Wavertree .
Donations of £1 to the appeal can be made by texting 'PJTF 68' to 70070.
For more information, go to www.facebook.com/rosefarleyappeal
Wavertree
Mark Womack
Childwall
West Derby
RuncornTributes paid to man killed in tragic crash between lorry and two cars"Very sad loss of a young life"


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text:
An Enemy of the People – Gate Theatre – Review
By No More Workhorse on May 28, 2013 • ( 6 Comments )
The development of Kirsten Springs promises to change the lives of all those in its locality. New wealth has come to the area and land prices are starting to increase rapidly. It is all due to the new spa and the healing powers of the local waters. Dr. Thomas Stockmann is an idealist and a friend of many of the radicals and free thinkers in the town. When he makes the discovery that the water in the lake is polluted with waste from the local tannery, he naively thinks he will be hailed as a hero.
Central to the play is the battle between Dr. Thomas Stockmann (Declan Conlon) and the Mayor of the town, his brother, Peter Stockmann (Denis Conway). It is clear from the outset that there have always been differences between the two in their lifestyle and views. The play is based on one man's attempt to tell the truth, while the other tries to stop the truth coming out at all costs.
This play was written by Norwegian playwright Henrik Ibsen in 1882 and adapted by Arthur Miller in 1950. Ibsen wrote it after the outcry at his play 'Ghosts' which was considered shocking at the time it was written due to its portrayal of hypocrisy in Victorian morality. The central theme of the play is people's unwillingness to accept the truth, if a lie is easier and better financially. There are obvious comparisons between these times and modern Ireland, and the play seems very apt for the world we live in.
This is an impressive new production of the Ibsen play that breathes new life and imagination into it. The scene with the town meeting, where Thomas tries to tell the people what he has discovered converts the gate theatre into a meeting hall, with many of the actors shouting comments and criticisms at the stage from amongst the audience. This scene is central to the production and serves to bring the audience into the performance.
The play deals well with the moral dilemma of a man who has the choice to stand by his convictions or fold to popular opinion. There are several scenes that feel a bit long, and the final scene is a little dragged out, but these small faults are outweighed by the many positives in the play. The direction of Wayne Jordan is adventurous and entertaining. The battle between the two brothers is fascinating and the performances of Conway and Conlon stand out. This is one of the best new productions I've seen in the Gate in recent times and is well worthy of your attention.
An Enemy of the People by Henrik Ibsen (Adapted by Arthur Miller) opens tonight at the Gate theatre (review was on last preview) and runs until July 13th, with tickets from €25. Matinees 8th June, 6th and 13th July all priced at €25.
Cast includes:
Fiona Bell, Liam Carney, Steve Cash, Declan Conlon, Denis Conway, Siobhan Cullen, Robert Duff, Jill Harding, Bosco Hogan, Mark Huberman, Ronan Leahy, Callum Martin, Barry McGovern, Morgan Moore, Peter O'Byrne, Donncha O'Dea, James O'Donoghue
Directed by Wayne Jordan
Set Design by Paul O'Mahony
Costume Design by Joan O'Clery
Lighting Design by Davy Cunningham
Music and Sound Design by Philip Stewart
Categories: Theatre
Things to do in Dublin this week (27-05-13 to 2-06-13)
Forbidden Fruit Night
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text:
\section{Introduction}
Given a collection of sequences, each being a multidimensional time series produced by the same latent mechanism, we consider learning this mechanism and predicting a sequence's future evolution. More precisely, suppose there is an unknown (possibly highly nonlinear) map $\phi$ that evolves any initial condition in discrete time $i$ according to
\[
    x_{i+1} = \phi(x_i).
\]
Provided with training data $\{x_{i,j}\}$, where each $j \in [M]$ corresponds to such a sequence ($i \in \{0,\ldots,N_j\}$), we'd like to learn $\phi$ purely from the data, or more precisely an approximation of its image $\tilde{\phi}(x) \approx \phi(x)$ for any $x$ in the problem domain. This way, one can for example perform continuation of existing sequences via $\tilde{x}_{i+1,j} = \tilde{\phi}(\tilde{x}_{i,j})$ for $i \geq N_j$ and $\tilde{x}_{N_j,j}=x_{N_j,j}$, or predict a sequence evolved from a new initial condition via $y_{i+1} = \tilde{\phi} (y_i)$ for $i>0$.

This task of course appears in many contexts (see e.g., Sec.\ref{sec:relatedWorks}). This article considers a very specific one, for which the latent map $\phi$ is assumed to be \emph{symplectic}: for simplicity we will work with finite dim. vector spaces equipped with canonical symplectic structure, which means each $x$ can be written as $x=[\bm{p}, \bm{q}]$ where $\bm{p}, \bm{q} \in \mathbb{R}^d$, and the Jacobian of $\phi$ satisfies
\[
    (\phi')^T J \phi' = J,
\]
where $J=\begin{bmatrix} 0 & I \\ -I & 0 \end{bmatrix}$ is a $2d$-by-$2d$ matrix with $0$ and $I$ being $d$-by-$d$ blocks.

The consideration of symplectic evolution maps is largely motivated by the learning and prediction of mechanical behaviors, which recently attracted significant attention  (see Sec.\ref{sec:relatedWorks} 3rd paragraph). More precisely, if the latent evolution mechanism is provided by a Hamiltonian mechanical system, each time series is given by a solution to the Hamiltonian ODE system sampled at discretized time points. That is, $x_i=[\bm p_i,\bm q_i]$, $\bm p_i = \bm p(ih), \bm q_i = \bm q(ih)$, with
\begin{equation}
    \dot{\bm p}(t) = -\frac{\partial H}{\partial \bm q} (\bm p(t),\bm q(t)),\,
    \dot{\bm q}(t) = \frac{\partial H}{\partial \bm p} (\bm p(t),\bm q(t)), \,
    \label{eq:HamiltonianODE}
\end{equation}
where $H(\cdot,\cdot)$ is the latent Hamiltonian function and $h>0$ is the sampling time step. In this case, the latent $\phi$ we're trying to learn is the flow map of ODE \eqref{eq:HamiltonianODE}, defined as
\[
    \phi [\bm p(t), \bm q(t)] := [\bm p(t+h), \bm q(t+h)],    \quad \forall t.
\]
Given any $h>0$, the corresponding $\phi$ is a symplectic map (e.g., \citealp{goldstein1980classical}), and that is why the learning of symplectic maps is relevant.

Worth noting is, a popular and successful line of thoughts is based on learning the right hand side of the latent ODE, which in our case (eq.\ref{eq:HamiltonianODE}) amounts to either generic approaches that learn the vector-valued function $f(\bm{p}, \bm{q}) := [-\partial H/\partial \bm q, \partial H/\partial \bm p]$ (see Sec.\ref{sec:relatedWorks} 1st paragraph), or specialized methods that utilize the problem structure and learn the scalar-valued function $H(\bm{p}, \bm{q})$ (see Sec.\ref{sec:relatedWorks} 3rd paragraph). This article, however, is based on a different idea, namely to directly \textbf{learn the evolution map} $\phi$.

The advantages of doing so include: (i) \underline{Generality}: it works no matter whether there is an underlying ODE system (see Sec.\ref{sec:Results:standardMap} for an example where there isn't). (ii) \underline{Local Accuracy}: for purely data driven problems, learning the map has \emph{one} source of error, namely the approximation error of the map, whereas learning vector field generally has \emph{three}: first one has to estimate the vector field from data, for example by finite difference which incurs error, then there is approximation error of the vector field, and finally the vector field needs to be numerically integrated in order to make predictions and this creates error too. (iii) \underline{Symplecticity (and Global Accuracy)}: we will propose a simple way to exactly maintain the symplecticity of $\phi$ despite using an approximation\footnote{Vector-field-based methods can also be designed to make symplectic predictions; see Sec.\ref{sec:relatedWorks} 4th paragraph.}; we will also rigorously show, when considering how local error accumulates for making long time predictions, exact symplecticity can help significantly by keeping the error accumulation additive, so that global error grows linearly instead of exponentially.

\begin{example}
    To make things concrete, consider latent Hamiltonian dynamics $\dot{x}=Ax$ where $A=-A^T$. A vector-field-based method aims at learning the function $Ax$, but if discrete time-series are the only available data, it actually learns $\tilde{A}x$ instead, where  $\tilde{A}=(\exp(Ah)-I)/h$ if 1st-order finite difference is used for estimating the vector-field. A map-based method seeks the map $\exp(A h)x$ instead. When later making predictions, the vector-field-based method numerically integrates the vector field $\tilde{A}x$, which oftentimes corresponds to constructing a polynomial in $h$ approximation of $\exp(\tilde{A} h)$; on the contrary, a map-based method requires no numerical integration. A similar comparison holds for nonlinear cases too.
\end{example}

In order to learn a \emph{symplectic} evolution map, we use a tool known as generating functions, which have one-to-one correspondence with symplectic maps. We use a Neural Network, however not for approximating the latent symplectic map, but to approximate its corresponding Generating Function (the method is thus called GFNN). By doing so, the associated evolution map is always symplectic, whether or not it is a good approximation of the latent evolution map, and an appropriate neural network, even just a feedforward one, will be a good approximation after training (see Rmk.\ref{rmk:universalApprox}). This symplectic map representation is intrinsic, purely due to the symplectic structure, and no regularization is used.

Moreover, the guaranteed symplecticity originated from  the generating function technique allows us to obtain a nontrivial, linearly growing bound on the prediction error:

\begin{theorem}[Informal version of Thm.\ref{thm:main}]
Consider latent dynamics far from chaos (more precisely, being integrable). If the latent generating function is approximated with $\leq \varepsilon$ error in first derivatives, then except for a set of initial conditions whose measure goes to 0 as $\varepsilon \downarrow 0$, the deviation between the predicted sequence $\left( \qpt{0} \right), \left( \qpt{1}, \right), \ldots$ and the true sequence satisfies
    \begin{align*}
    \left\{
    \begin{aligned}
    \norm{\bm p_n - \bm p\left( n h \right)}_2
         \le C \cdot \left( n h \right) \cdot \varepsilon, \\
    \norm{\bm q_n - \bm q\left( n h \right)}_2
         \le C \cdot \left( n h \right) \cdot \varepsilon, \\
    \end{aligned}
    \right.
    \quad \forall n \le h^{-1} \varepsilon^{-1},
    \end{align*}
    for some constant $C>0$, where $n$ is the number of prediction steps and $h$ is the sampling time step of the data.
\end{theorem}
The merit of this bound lies in long time predictions: note $n$ can be arbitrarily large as $\varepsilon$ can be infinitesimal ($h$ is fixed by the training data, and $nh$ is the physical prediction time).

\textbf{A brief summary of main contributions:}

$\bullet$ (Algorithm) \emph{Learn map instead of vector field. Exact symplecticity guaranteed by generating function representation.}

$\bullet$ (Theory) \emph{Linear bound on long-time prediction error.}

$\bullet$ (Validation) \emph{Systematic empirical investigations.}

\subsection{Related Works and Discussion}
\label{sec:relatedWorks}

Learning and then predicting dynamics from data is an extremely active research direction. While it is impossible to review all important works, we first mention the classical area of time series (e.g., \citealp{box2015time, abarbanel2012analysis, kantz2004nonlinear, bradley2015nonlinear}), where latent differential equations may or may not be assumed. For cases where a latent differential equation is believed to exist, which may correspond to a complex and/or un-modeled underlying dynamical process, some seminal works include \cite{baake1992fitting, bongard2007automated, schmidt2009distilling},
and more recent progress include
those based on learning (part of) the vector field via sparse regression of a library (e.g., \citealp{brunton2016discovering, tran2017exact, schaeffer2018extracting, lu2019nonparametric, rudy2017data, kang2019ident, reinbold2021robust}),
learning the vector field via neural network (e.g., \citealp{raissi2018multistep, rudy2019deep, qin2019data, long2018pde}),
and learning the vector field via other approaches such as Gaussian processes (e.g., \citealp{raissi2018hidden}).

`Model-free' approaches that are based on machine learning techniques for sequences have also been proposed, such as
\cite{bailer1998recurrent} (vanilla RNN),
\cite{wang2017new} (LSTM),
\cite{pathak2018model} (reservoir computing),
\cite{mukhopadhyay2020learning} (CNN), and
\cite{shalova2020tensorized} (transformer).

Faced with the extreme success of these generic methods, interests have also been growing in incorporating domain knowledge and specific structures of the underlying problems into the otherwise-black-box schemes (see e.g., \citealp{raissi2019physics}).
In terms of mechanical problems modeled by Hamiltonian systems, seminal progress include HNN \cite{greydanus2019hamiltonian} and an independent work \cite{bertalan2019learning}, SRNN \cite{chen2019symplectic}, SympNets \cite{jin2020sympnets}, and \cite{lutter2018deep,toth2019hamiltonian,
zhong2019symplectic,
wu2020structure
xiong2021nonseparable}, all of which, except SympNets, are related to learning some quantity that produces the Hamiltonian vector field.

In particular, both HNN and SRNN are based on the great idea of learning (using a neural network) the Hamiltonian that generates the vector field (VF), instead of learning the VF itself; this improves accuracy as the Hamiltonian structure of the VF will not be lost due to approximation.
HNN learns the Hamiltonian by matching its induced VF with the latent VF (when such information is unavailable, for example in a purely data driven context, data-based approximation such as finite-difference is needed). Then it predicts by numerically integrating the learned VF, and for this we note a Hamiltonian VF doesn't guarantee the symplecticity of its integration\footnote{Unless a symplectic integrator is used. Note the seminal work of HNN used RK45 which is not symplectic, however with small error tolerance (thus good precision but high computation cost).}.
SRNN, on the other hand, learns the Hamiltonian by matching its symplectic integration with the training sequences, and its prediction is then given by symplectic integration of the learned Hamiltonian. It is therefore the closer to GFNN as it essentially learns a symplectic map; it is just that SRNN represents this map by a symplectic integration of a neural-network-approximated Hamiltonian, whereas GFNN represents it by a neural-network-approximated generating function. Because of this, SRNN doesn't need finite-difference approximation and has good prediction accuracy, but it only works for symplectic maps originated from Hamiltonian ODEs, and its accuracy is hampered if the latent Hamiltonian is nonseparable\footnote{The original SRNN is based on symplectic integrators for separable Hamiltonians, and nonsymplectic integrators for nonseparable ones; see Footnote \ref{ftnt:SRNN_nonseparable} for additional information.}.

In comparison, GFNN is not based on Hamiltonian vector fields. It is purely data driven, always symplectic, and works the same for separable-, nonseparable-, or even non-Hamiltonian latent systems.

Worth mentioning is the clever recent work of SympNets \cite{jin2020sympnets}, which also enjoys most of the aforementioned qualitative features of {GFNN}{}. It complements {GFNN}{} and echoes with our view that directly approximating symplectic maps (instead of Hamiltonian vector fields) in an exactly symplectic way is advantageous (note SRNN can also be seen as a (different) way of doing so). \emph{Algorithmically}, SympNets stack up triangular maps (inspired by symplectic integrator) to construct specialized (new) neural networks, which represent only symplectic maps, and then use them to directly approximate the latent evolution map; GFNN on the other hand uses generating function to indirectly represent the evolution map, and because of its mathematical structure, exact symplecticity is automatically guaranteed, and no special neural network is needed for representing the generating function. Consequently, the \emph{theory} of SympNets is devoted to a universal approximation theorem that characterizes the local prediction error, whereas we focus on the global prediction error (i.e., error after many steps of prediction, instead of one) and rigorously show a nontrivial fact that local errors only accumulate linearly into global error; no approximation theorem needed as it's already established for generic networks. In terms of \emph{performance}, we observe SympNets outperforming vector-field-based approaches (as reasoned above), but GFNN has further improved performance; see e.g., \cref{fig:2body}, for which we tried up to 30 layers with 10 sublayers using SympNets' code (both LA- and G-SympNets) and plotted its best result, namely LA-SympNets with 30 layers and 10 sublayers (c.f., here {GFNN}{} used 5 layers). We feel SympNets generally require a significantly deeper network than {GFNN}{} to achieve high approximation power, but then training and computational challenges may arise.

One more remark is, a good amount of existing work considered predicting chaotic dynamics, but a major part of this work is concerned with structured, non-chaotic dynamics, for which controlled long time (strong) accuracy in individual trajectories becomes possible. Predicting chaos is nontrivial, but sometimes the existence of a chaotic attractor makes the system forgiving, and because of that, prediction errors do not accumulate as much as they can in non-chaotic systems. Besides, one often cares more about statistical accuracy for chaotic systems (e.g., \citealp{tsai2020learning}), as opposed to strong accuracy in trajectory (which usually grows too fast in chaos; see Rmk.\ref{rmk:chaosPositiveLyapunov}). Meanwhile, accurately predicting the trajectory of non-chaotic systems is desirable in numerous applications. Nevertheless, {GFNN}'s predictive power for chaos will be empirically confirmed too.

\section{Methods}

\subsection{Symplectic Map and Generating Function}
As we do not assume or seek a latent ODE system but directly approximate the evolution map, a representation of this symplectic map is essential. Instead of directly approximating it, which has the extra difficulty of losing symplecticity (which has to be exact), we use a mathematical tool known as generating function. Let us be more specific:

Firstly, given a (type-2) generating function, there is an associated symplectic map (a.k.a. canonical transformation):

\begin{lemma}
    Consider a differentiable function $F(\bm q, \bm P)$ which shall be called a generating function. The map $[\bm p, \bm q] \mapsto [\bm P,\bm Q]$ implicitly defined by
    $
        \bm p = \frac{\partial F}{\partial \bm q}(\bm q,\bm P)$,
    $
        \bm Q = \frac{\partial F}{\partial \bm P}(\bm q,\bm P),
    $
    is a symplectic map.
\end{lemma}

\begin{proof}
    See e.g., \cite{goldstein1980classical}.
\end{proof}

The converse is also true, as long as the latent map doesn't correspond to an evolution time too long (otherwise singularities can be developed):

\begin{lemma}
    For any infinitesimal symplectomorphism (i.e., symplectic map) on $T^*\mathbb{R}^d$ (i.e., vector phase space), there is a corresponding generating function.

\end{lemma}
\begin{proof}
    This is because the first cohomology group of $T^*\mathbb{R}^d$ is trivial; see e.g., \cite{da2001lectures}.
\end{proof}

\begin{remark}[generating functions and Hamiltonian system]
    We do not assume the latent map that generates the data in discrete time corresponds to an underlying Hamiltonian ODE system in continuous time. There are symplectic maps that do not have such correspondence (see e.g., Sec.\ref{sec:Results:standardMap}).

    On the other hand, given a Hamiltonian system, its flow map, defined as $\phi^t : [\bm p(0), \bm q(0)] \mapsto [\bm p(t), \bm q(t)]$, is symplectic for any $t$. Therefore, there is a family of corresponding generating functions $F(\bm q, \bm P, t)$, each of which generates the symplectic map $[\bm P, \bm Q]=\phi^t [\bm p, \bm q]$. Moreover, the relation between the Hamiltonian $H$ and $F$ can be made more direct via the Hamilton-Jacobi PDE:
    $
        H\left(\frac{\partial F}{\partial \bm q}, \bm q, t\right) + \frac{\partial F}{\partial t} = 0.
    $
\end{remark}

Because of their 1-to-1 correspondence, instead of approximating the symplectic evolution map $\phi:[\bm p,\bm q]\mapsto[\bm P,\bm Q]$, we use a Feedforward Neural Network to approximate the corresponding generating function $F(\bm q,\bm P)$. This way, no matter how much error the FNN has in approximating $F$, it always gives to an evolution map that is \emph{exactly} symplectic.

\subsection{Learning Based on Generating Function Training}
The type-2 generating function corresponding to a $h$-time flow map is $F\left( \bm q, \bm P \right)=\bm q \cdot \bm P + \mathcal{O}(h)$, and what varies across different problems is inside the $\mathcal{O}(h)$ term. Therefore, for easier training we learn an equivalent, modified generating function $S_h$, defined through $F \left( \bm q, \bm P \right) = \bm q \cdot \bm P + h \cdot S_h \left( \bm q, \bm P \right)$. It generates a sequence via iteration
\begin{align}\label{eq:Sh_seq}
    \left\{
    \begin{aligned}
        \bm p_{i} = \bm p_{i+1} + h \cdot \partial_1 S_h \left( \bm q_{i}, \bm p_{i+1} \right), \\
        \bm q_{i+1} = \bm q_{i} + h \cdot \partial_2 S_h \left( \bm q_{i}, \bm p_{i+1} \right), \\
    \end{aligned}
\right.
\end{align}
as long as an initial condition $[\qpt{0}]$ is provided.

To learn the latent $S_h$, GFNN uses a neural-network approximation $S_h^\theta$, and trains for a good parameterization $\theta$ to best satisfy \eqref{eq:Sh_seq}. See \cref{algorithm:1}.
\begin{algorithm}[ht]
 \caption{{GFNN}{}}\label{algorithm:1}
\begin{algorithmic}
\STATE {\bfseries{Data:}}
The data set $\left\{ {\left({\left[ \qpt{i, j} \right]}_{i=0}^{N_j}\right)}_{j=1}^M  \right\}$ is observed from sequences generated by a symplectic map $\phi^h$, with $\left[ \qpt{i, j} \right] \in \mathcal{D} \subseteq \mathbb{R}^d \times \mathbb{R}^d \cong T^* \mathbb{R}^d$ and $\qpt{i+1, j} = \phi^h \left( \qpt{i, j} \right)$.
\STATE {\bfseries{Training:}}
Optimize the loss function
\begin{align}\label{eq:loss}
    \begin{split}
        & \mathcal{L}_{GFNN} = \frac{1}{\sum_{j=1}^M N_j} \sum_{j=1}^M \sum_{i=0}^{N_j-1}\\
        & \quad \Big( \norm{h \partial_2 S_h^\theta(\bm q_{i, j}, \bm p_{i+1, j})-(\bm q_{i+1, j} - \bm q_{i, j})}_2^2 \\
    & \quad +\norm{h \partial_1 {S}_h^\theta(\bm q_{i, j}, \bm p_{i+1, j})-(\bm p_{i, j} - \bm p_{i+1, j})}_2^2 \Big).
    \end{split}
\end{align}
with respect to neural network parameters $\theta$ (see Appendix for our experimental details).
\STATE {\bfseries{Prediction:}}
Given any initial condition $(\bm q_0, \bm p_0) \in \mathcal{D}$,
one step evolution to $(\tilde{\bm q}_1, \tilde{\bm p}_1)$ can be solved from
\begin{align}\label{eq:prediction}
    \left\{
    \begin{aligned}
        \bm p_0 = \tilde{\bm p}_1 + h \cdot \partial_1 S_h^\theta \left( \bm q_0, \tilde{\bm p}_1 \right), \\
        \tilde{\bm q}_1 = \bm q_0 + h \cdot \partial_2 S_h^\theta \left( \bm q_0, \tilde{\bm p}_1 \right). \\
    \end{aligned}
\right.
\end{align}
This can be iterated.
\end{algorithmic}
\end{algorithm}

\section{Global Error Analysis}\label{sec:error_analysis}

We now show that, under reasonable assumptions, {GFNN}{}'s prediction will be close to the true sequence (continued by the latent $\phi$) for a very long time, as a linearly growing long time error bound will be established. This will be contrasted with an obtainable exponentially growing error bound for generic vector-field-based methods. The latter methods are of course more versatile but they do not utilize the special symplectic structure. Proofs are based on normal form and KAM-type techniques and deferred to Appendix.

The main condition needed for this mild error growth is integrability, which, very roughly speaking, requires the latent system to be far from chaos, but it could still be highly nonlinear; see e.g., \cite{arnol2013mathematical}. In order to make it precise, some mathematical preparations are needed, but one can jump to Thm.\ref{thm:main} for the main results if preferred.

\begin{definition} A function $g(\bm{p}, \bm{q})$ is called a 1st-integral or a constant of motion of the dynamics if it remains constant as $\bm{p}, \bm{q}$ evolves in (continuous or discrete) time.
\end{definition}

\begin{definition}
    The (canonical) Poisson bracket of two arbitrary functions $f(\bm{p}, \bm{q}), g(\bm{p}, \bm{q})$ is another function defined as $\{f,g\}:=\langle \partial f/\partial q, \partial g/\partial p \rangle - \langle \partial f/\partial p, \partial g/\partial q \rangle$.
\end{definition}

\begin{theorem}[Arnold-Liouville]
Consider a $d$-degree-of-freedom Hamiltonian system. Assume there exist d independent $1$st integrals in the sense that the Poisson bracket of every pair is $0$. If the $d$-dim. surfaces implicitly defined by the level sets of those $1$st integrals are compact, then there exists a canonical transformation from $\bm{p}, \bm{q}$ to $\bm I, \bm \varphi$, such that $\bm \varphi$ can be defined on the $d$-torus, and in the new variables the Hamiltonian only depends on $\bm I$. In this case, $\bm I$, $\bm \varphi$, and the Hamiltonian are respectively called the action, angle variables, and an integrable Hamiltonian.
\end{theorem}
\begin{proof} See e.g., \cite{arnol2013mathematical}.
\end{proof}

\begin{remark}
In an integrable system, the action variables $\bm I$ remain constants (they are canonical versions of the first integrals), while the angle variables $\bm \varphi$ evolve on an invariant torus $\left\{ \bm I=\bm I\left(0\right), \bm \varphi \in \mathbb{T}^d \right\}$, where
\begin{align*}
    \mathbb{T}^d & =\mathbb{R}^d\slash (2\pi\mathbb{Z}^d)
    =\left\{ \left( \varphi_1, \ldots, \varphi_d \right) \Mod 2\pi; \varphi_i \in \mathbb{R} \right\}.
\end{align*}
It is easy to show that the fixed time step generating function for an integrable system takes the form of $S_h \left( \bm I_0, \bm \varphi_1 \right) = H\left( \bm I_0 \right)$, as the exact time-h flow is defined as the following
\begin{align}\label{eq:Iphi_dynamics}
    \left\{
    \begin{aligned}
        \bm I_{1} & = \bm I_{0}, \\
        \bm \varphi_{1} & = \bm \varphi_{0} + h \, \partial_{1} S_h \left( \bm I_{0}, \bm \varphi_1 \right) = \bm \varphi_{0} + h \, \nabla H \left( \bm I_0 \right). \\
    \end{aligned}
    \right.
\end{align}
Denote $\nabla H \left( \bm I \right)$ by $\bm \omega \left( \bm I \right) = \left[ \omega_1 \left( \bm I \right), \ldots, \omega_d \left( \bm I \right) \right]$.
It can be directly seen from \cref{eq:Iphi_dynamics} that $\omega_i \left( \bm I \right)$ represents the change rate (i.e., frequency) of the angle variable $\varphi_i$.
\label{rmk_invariantTori}
\end{remark}

Our theory works for almost all initial frequencies, and to describe what are the exceptions we need the following definition, which generalizes irrational numbers in some sense.

\begin{definition}[$\left(\gamma, \nu\right)$-Diophantine condition\footnote{also known as strong non-resonance condition}]
    Frequency vector $\bm \omega = \left\{ \omega_1, \omega_2, \ldots, \omega_d \right\}$ satisfies $\left( \gamma, \nu \right)$-Diophantine condition iff
    $\,\left| \bm k \cdot \bm \omega \right| \ge \gamma \cdot {\norm{\bm k}_1}^{-\nu}$, $\forall \bm k \in \mathbb{Z}^d$, $\bm k \neq \bm 0$,

    for some $\gamma>0, \nu>0$.
\label{def:Diophantine}
\end{definition}

\begin{definition}[$\varepsilon$-neighborhood condition]
$\left( \bm{p}, \bm{q} \right) \in \mathbb{R}^d \times \mathbb{R}^d$ of an integrable system satisfies $\varepsilon$-neighborhood condition if there exists $\bm I^* \in \mathbb{R}^d$,
such that $\bm \omega\left( \bm I^* \right)$ satisfies the $\left( \gamma, \nu \right)$-Diophantine condition (Def.\ref{def:Diophantine}),
and $\norm{\bm I\left( \bm{p}, \bm{q} \right) - \bm I^*}_2 \le c \cdot {\left| \log{\varepsilon} \right|}^{-\nu-1}$ for some $\varepsilon$ independent constant $c$ (defined in the Appendix) with
$\bm I \left( \bm{p}, \bm{q} \right)$ being the actions of the system.
\label{def:epsNbhCond}
\end{definition}

With these preparations, we see the action and angle variables $\bm{I}, \bm{\varphi}$ form a new coordinate system alternative to $\bm{p}, \bm{q}$ (note even if the system is not integrable and/or time is no longer continuous, one is still free to perform any canonical coordinate transformation; it's just doing so may or may not reveal structured dynamics any more). In fact, they give finer estimates of the prediction error:

\begin{theorem}[GFNN's long-time prediction error in actions and angles]\label{thm:GFNN_action_angle}
    Consider an integrable Hamiltonian system written in action-angle variables,
    whose exact time-h flow map corresponds to generating function $S_h(\cdot, \cdot)$.
    Predict its trajectory using {GFNN}{} with learned generating function $S_h^\theta \left( \cdot, \cdot \right)$ in a bounded data domain $\mathcal{D} = \mathcal{D}_1 \times \mathbb{T}^d \subseteq \mathbb{R}^d \times \mathbb{T}^d$.
    $\exists\,\varepsilon>0, \rho>0$, such that if the learned generating function ${S}_h^\theta$ (extended in a complex neighborhood of $\mathcal{D}$) is analytic and satisfies
        \begin{align*}
            \sum_{i=1,2} \norm{\partial_i S_h^\theta \left( \cdot,\cdot \right) - \partial_i S_h \left( \cdot,\cdot \right)}_{\infty} \le C_1 \varepsilon,
        \end{align*}
    for some $\varepsilon$ independent constant $C_1$,
    where the $L^\infty$ norm is defined over the $\varepsilon$ independent complex neighborhood $\mathcal{B}_\rho\left(\mathcal{D}\right)$ of $\mathcal{D}$,
    then, $\forall \left( \Iphit{0} \right) \in \mathcal{D}$ that satisfies $\varepsilon$-neighborhood condition (Def.\ref{def:epsNbhCond}), the predicted sequence
     $\left( \Iphit{0} \right), \left( \Iphit{1}, \right), \ldots$ generated by {GFNN}{} satisfies
    \begin{align}
    \left\{
        \begin{aligned}
        & \norm{\bm I_n -  \bm I(0)}_2 \le C \cdot \varepsilon, \\
        & \norm{\bm \varphi_n -  \bm \varphi(n h)}_2 \le C \cdot \left( n h \right) \cdot \varepsilon, \\
        \end{aligned}
    \right.
    \quad \forall n \le h^{-1} \varepsilon^{-1},
    \end{align}
    for some constant $C$.
\end{theorem}

The intuition behind the proof of \cref{thm:GFNN_action_angle} (which is in Appendix) is the following: the predicted dynamics $(\bm I_n, \bm \varphi_n)$ and the true dynamics $(\bm I(nh), \bm \varphi(nh))$ deviate because each step of the prediction introduces some error due to inaccurate $S_h^\theta$, but these errors accumulate in a very delicate way; in fact, earlier errors cannot be amplified too much in order for a linear bound to exist. The key reason, as the proof will recover, is that $\bm I_n$'s dynamics is mostly just oscillatory in time. We show this by decomposing the predicted dynamics into a macroscopic part plus microscopic oscillations. The macroscopic part can be proved to correspond to a barely changing action. The microscopic part, on the other hand, does not accumulate.

Specifically, we introduce a carefully-chosen near-identity canonical coordinate change $\mathcal{T} : \left[ \bm{I}, \bm{\varphi} \right] \mapsto \left[\bm{J}, \bm{\theta} \right]$, $\mathcal{T} \approx id+\mathcal{O}(\varepsilon)$, and show that the new variables $[\bm J_n, \bm\theta_n]$ describe, roughly, the macroscopic part of the predicted dynamics.

We then prove, when compared to the true dynamics,
    \begin{align*}
    \left\{
    \begin{aligned}
    & \norm{\bm I(n h) - \bm J_n}_2 = \mathcal{O}\left( \epsilon \right),\\
    & \norm{\bm \varphi (n h) - \bm \theta_n}_2 = n h \cdot \mathcal{O} (\varepsilon), \\
    \end{aligned}
    \right.
    \quad \forall nh = \mathcal{O}\left( \varepsilon^{-1} \right).
    \end{align*}
Since $\mathcal{T}$ is near-identity, $[\bm J_n,\bm\theta_n] = [\bm I_n,\bm\varphi_n]+\mathcal{O}(\varepsilon)$ for all $n$, and the triangle inequality then completes the proof. $\square$

\begin{figure}[ht]
\vskip 0.2 in
\begin{center}
\begin{tikzcd}[nodes in empty cells, column sep=1.2em,row sep=2.0em]
    & \left[ \Iphit{n} \right] \arrow[rr, leftrightarrow, "\clap{canonical}"] \arrow[d, leftrightarrow, "\substack{\rlap{same set }\\\rlap{of  variables}}"] &  & \left[ \qpt{n} \right], S_h^\theta \arrow[d, "\substack{\clap{same set of variables}\\\clap{different dynamics} }", leftrightarrow] \\
    & \substack{\left[ \bm I(t), \bm \varphi(t) \right] \in \\ \clap{Invariant Torus}\\\left\{\bm I=\bm I(0),\,\bm \varphi \in \mathbb{T}^d\right\}} \arrow[rr, leftrightarrow, "\clap{canonical}"] &  & \substack{\clap{exact }\\\left[ \bm p(t), \bm q(t) \right]} \\
    \exists \left[ \Jthetat{n} \right] \arrow[ur, bend right, leftrightarrow, "\substack{

    \rlap{dynamics: $\mathcal{O}\left( \varepsilon \right)$ close till $t=\mathcal{O}\left( \varepsilon^{-1} \right)$}}"]

    \arrow[uur, bend left, leftrightarrow, "\substack{\clap{canonical $\mathcal{T}$}}"] & & &
\end{tikzcd}
\caption{Main components in the proof of linear error growth.
\label{proof_diagram}
\end{center}
\vskip -0.2 in
\end{figure}

Now we can relate the error bound in  \cref{thm:GFNN_action_angle} back to that for the original variables $\bm{p}, \bm{q}$. The big picture is summarized by Fig.\ref{proof_diagram}. In the end, our theory only requires the existence of action and angle variables, and no knowledge about how to find the actions and angles is needed.

\begin{theorem}[linear growth of GFNN long-time prediction error]
    Consider an integrable Hamiltonian system whose exact solution is denoted by $\bm p(t), \bm q(t) \in \mathbb{R}^d$. Denote by $S_h \left( \cdot, \cdot \right)$ the generating function corresponding to its exact time-$h$ flow map. Consider predicting its trajectory using {GFNN}{} with learned generating function $S_h^\theta \left( \cdot, \cdot \right)$ in a bounded data domain $\mathcal{D} \subseteq \mathbb{R}^d \times \mathbb{R}^d$.
    $\exists\,\varepsilon>0, \rho>0$, such that if the learned generating function ${S}_h^\theta$ (extended in a complex neighborhood of $\mathcal{D}$) is analytic and satisfies
        \begin{align}
            \sum_{i=1,2} \norm{\partial_i S_h^\theta \left( \cdot,\cdot \right) - \partial_i S_h \left( \cdot,\cdot \right)}_{\infty} \le C_1 \varepsilon,
            \label{eq:neededNNaccuracy}
        \end{align}

    for some $\varepsilon$ independent constant $C_1$,
    where the $L^\infty$ norm is defined over the $\varepsilon$ independent complex neighborhood $\mathcal{B}_\rho\left(\mathcal{D}\right)$ of $\mathcal{D}$,
    then, $\forall \left( \qpt{0} \right) \in \mathcal{D}$ that satisfies $\varepsilon$-neighborhood condition (Def.\ref{def:epsNbhCond}) with nonlinear frequency $\bm\omega(\cdot)$ being given by $S_h$,
    the predicted sequence
     $\left( \qpt{0} \right), \left( \qpt{1}, \right), \ldots$ generated by {GFNN}{} satisfies
    \begin{align}
    \left\{
    \begin{aligned}
    \norm{\bm p_n - \bm p\left( n h \right)}_2
         \le C \cdot \left( n h \right) \cdot \varepsilon, \\
    \norm{\bm q_n - \bm q\left( n h \right)}_2
         \le C \cdot \left( n h \right) \cdot \varepsilon, \\
    \end{aligned}
    \right.
    \quad \forall n \le h^{-1} \varepsilon^{-1},
    \end{align}
    for some constant $C>0$.
    \label{thm:main}
\end{theorem}

\begin{remark}
    It is known that neural networks can approximate functions and their derivatives with any precision; see e.g., the classical work \cite{hornik1990universal} and a more recent discussion \cite{yarotsky2017error}. \eqref{eq:neededNNaccuracy} can thus be attained.
    \label{rmk:universalApprox}
\end{remark}

\begin{remark}
    The integrability assumption in Thm.\ref{thm:main} is nontrivial, however reasonable. This is because it rules out the possibility of a positive Lyapunov exponent, which by definition indicates that a deviation between two trajectories can exponentially grow in time (e.g., \citealp{alligood1996chaos}). Naturally, if the latent system does have a positive Lyapunov exponent, then in general one should not expect a linearly growing prediction error, as an arbitrarily small approximation error, even if it's just made in one step, can be exponentially amplified.

    A simple illustration of this is a Hamiltonian system $\dot{x}=y, \dot{y}=x$, which is not integrable due to noncompactness (not even chaos). It has a Lyapunov exponent of $+1$. Consider predictions based on approximation $\dot{x}=y+\delta_x, \dot{y}=x+\delta_y$, then no matter how small $\delta_x$ and $\delta_y$ are, the difference between its solution and the original one grows like $\exp(t)$ except for measure zero $\delta_x$ and $\delta_y$ values.
    \label{rmk:chaosPositiveLyapunov}
\end{remark}

As a comparison, if the prediction map is not symplectic, either due to nonsymplectic numerical integration, or because the learned vector field is no longer Hamiltonian, local prediction error (in each step) may get amplified and long time prediction error may grow exponentially:

\begin{theorem}
	Consider the latent dynamics $\dot{x}=f(x)$ and its prediction via an Euler integration of the learned vector field $x_{i+1}=x_i+h \tilde{f}(x_i)$, with consistent initial condition $x(0)=x_0$. Assume $f$ is $L$-Lipschitz continuous, $\mathcal{C}^1$, the learned vector field is accurate up to $\delta$ in the sense that $\|\tilde{f}-f\|_\infty \leq \delta$, and the prediction remains bounded. Then the accuracy of the prediction at time $T=nh$ satisfies
	\[
		\|x(T)-x_n\| \leq \frac{\exp(L T)-1}{L}(\delta+L h/2).
	\]
	\label{thm:genericPredictionErrorBound}
\end{theorem}

\begin{proof}
See Appendix.
\end{proof}

\begin{remark}
	This exponential growth with $T$ (and $n$) is not an overestimation. A simple example that attains it is $f(x)=x$ and $\tilde{f}(x)=x+\delta$. This is of course because the latent dynamics is structurally bad and does not forgive past errors, but that is exactly our point: when the latent dynamics has specific structures such as being a symplectic flow, utilizing those structures in the prediction could lead to much better controlled accumulation of errors.
\end{remark}

\begin{remark}
In the context of learning dynamics from data, two sources contribute to the difference between $f$ and $\tilde{f}$. One is approximation error, for instance of the neural network; the other is because one doesn't have an oracle about the latent vector field $f$ but only its approximation from the data, for example $f(x_i)\approx (x_{i+1}-x_i)/h$. A map based approach doesn't directly use $f$ and thus can avoid the latter error, and it doesn't have numerical integration errors in the next phase of predictions either. A neural ODE type treatment \cite{zhong2019symplectic} can avoid the latter error too, but integration errors in the prediction phase remain (unless computationally expensive small steps are used).
\end{remark}

\section{Experiments}\label{sec:experiments}
Let's now systematically (within the page limit) investigate the empirical performances of GFNN. It was conjectured that invariant sets of a smooth map with a dense trajectory are typically either periodic, quasiperiodic\footnote{A function $f(t)$ is quasiperiodic if $\exists$ some constants $n\in\mathbb{Z}^+$, $\Omega_1,\cdots,\Omega_n\in \mathbb{R}$, and some function $F$ 1-periodic in each argument, s.t., $f(t)=F(\Omega_1 t,\cdots,\Omega_n t)$. An integrable system's solution is quasiperiodic if LCM$(\omega_1(\bm I), \cdots, \omega_d(\bm I))$ doesn't exist (see Rmk.\ref{rmk_invariantTori} for $\omega_i(\bm I)$); otherwise it is periodic.
\label{note:quasiperiodicity}}, or chaotic \cite{sander2015many}. Thus, Sec.\ref{sec:2body}-\ref{sec:Results:standardMap} will study classical examples that respectively correspond to periodic, quasiperiodic+chaotic, quasiperiodic, and quasiperiodic+chaotic cases. We'll see smaller and linearly growing errors of GFNN in both periodic and quasiperiodic cases, even when the latent system is not integrable. In chaotic cases, GFNN will also exhibit pleasant statistical accuracy.

VFNN stands for: learning the Vector Field via a Neural Network (without caring about the Hamiltonian structure).

Details of data preparation and training are in \cref{appendix:experimental_details}.

\subsection{An Integrable and Separable Hamiltonian: 2-Body Problem}
\label{sec:2body}

Consider the motion of 2 gravitationally interacting bodies. Letting their distance be $\bm q(t)$ and the corresponding momentum be $\bm p(t)$, the problem can be equivalently turned into (after unit normalization) an ODE system governed by
\[
 H\left( \bm{p}, \bm{q} \right) = \|\bm p\|_2^2/2 - 1/\norm{\bm q}_2.
\]
Despite its high nonlinearity, this is an integrable system. Analytical solutions known as Keplerian orbits exist and are periodic in bounded cases. Each solution is described by important physical quantities known as orbital elements, which include semi-major axis and eccentricity, that characterize the shape of the elliptic orbit.
As shown in \cref{fig:2body}, GFNN outperforms other methods and keeps the errors of semi-major axis and eccentricity small and bounded, which is consistent with \cref{thm:GFNN_action_angle} because semi-major axis and eccentricity are functions of actions known as Delaunay variables \cite{morbidelli2002modern}. The advantage of GFNN can also been seen in the original variables (e.g., $\bm q$), and the zoomed-in plots in row 2 show that the next two top performers are SRNN (seq\_len=2) and SRNN (seq\_len=5); SympNets has notably larger error in the orbital phase but its accuracy in the orbital shape is actually comparable to SRNN.

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
     \includegraphics[width=\linewidth]{cmp_2body.pdf}
     \vskip -10pt
     \caption{Comparison of $2$D Keplerian orbit predicted by different methods. The 1st row is the error growth of two variables of physical importance, namely semi-major axis and eccentricity (for this problem, their true values are both constants). Mean values of prediction errors starting from 1,000 i.i.d. initial conditions are plotted with shades corresponding to 1 standard deviation. The 2nd row zooms in the position variables of one predicted sequence (in $q_1$ and $q_2$ respectively). Data sequences are prepared with time step $0.1$.}\label{fig:2body}

\end{center}
\vskip -0.2in
\end{figure}

\subsection{A Non-integrable but Separable Hamiltonian System: H\'enon-Heiles}
\label{sec:HenonHeiles}

The H\'enon-Heiles system describes the motion of stars around a galactic center \cite{henon1964applicability}. It is a classical non-integrable system with very complex dynamics, governed by Hamiltonian
\[
H\left( p_1, p_2, q_1, q_2 \right)
= \frac{\left( p_1^2 + p_2^2 \right)}{2} + \left( \frac{q_1^2 + q_2^2}{2} + q_1^2 q_2 + \frac{q_2^3}{3}\right).
\]
Both chaotic and (quasi)-periodic solutions exist. Initial conditions corresponding to higher energy (i.e., $H$'s value) are more likely to be chaotic. We investigate {GFNN}'s performance in both cases.

\subsubsection{Non-periodic but regular motions}
For a non-chaotic initial condition, numerically observed was that the long time prediction error of GFNN still grows linearly even though the latent system is no longer integrable; see \cref{fig:regular_hh}. Notably, SRNN also exhibits linear error growth (although at a higher rate), and this is consistent with our intuition as SRNN also learns a symplectic evolution map (indirectly via the symplectic integration of a Hamiltonian to-be-learned). HNN, on the other hand, has exponentially growing error which quickly saturates to maximum values (due to boundedness of trajectories).

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
     \includegraphics[width=\linewidth]{cmp_hh_regular.pdf}
     \vskip -10pt
     \caption{Error of predictions of a quasi-periodic trajectory with energy level near $\frac{1}{12}$ of the H\'enon-Heiles system. Data sequences are prepared with time step $0.5$.}\label{fig:regular_hh}
 \end{center}
\vskip -0.2in
\end{figure}

{GFNN}'s linear error growth despite non-integrability is due to the existence of mathematical objects known as KAM-tori (e.g., \citealp{poschel1982integrability}). They correspond to part of the phase space where dynamics are topologically equivalent to integrable ones. A by-product is, solutions in this region are either quasiperiodic or periodic (see Footnote \ref{note:quasiperiodicity}).

\subsubsection{Dynamics in chaotic sea}
To visualize the prediction of chaotic dynamics, which take place in 4D, we use the standard tool of Poincar\'e section, which plots where an orbit intersects with a 2D slice of the 3D constant-energy manifold. \cref{fig:poincare_section_hh_chaotic} shows the Poincar\'e section produced by predictions of different methods, based on the same initial condition that leads to chaotic motion via the latent dynamics.
The true chaotic motion is ergodic on a submanifold of the phase space, and when restricting to the  Poincar\'e section, it gives intersections that are dense in a subset known as the chaotic sea. Therefore, the shapes of the dense area and the holes inside it (often corresponding to regular islands on which motions are (quasi)-periodic) are indicators of the prediction accuracy. Among methods tested in \cref{fig:poincare_section_hh_chaotic}, only VFNN didn't produce a pattern similar to the truth. Quantitative comparisons are conducted by comparing the empirical distributions of points on the Poincar\'e section, and KL divergences between their marginals and the truth are annotated along with the histograms. GFNN has the smallest errors.

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
     \includegraphics[width=\linewidth]{chaotic_hh2.png}
     \vskip -10pt
     \caption{Quantifying the statistical accuracy in predicting a chaotic orbit of the H\'enon-Heiles system. Left 3 columns: Poincar\'e section; right column: marginal distributions and their KL divergences from the truth. The plotted orbit corresponds to energy $\frac{1}{6}$; Poincar\'e section is given by $q_2(t)$, $p_2(t)$ at $q_1(t)=0$. Data sequences are prepared with time step $0.5$.}
     \label{fig:poincare_section_hh_chaotic}
\end{center}
\vskip -0.2in
\end{figure}

\subsection{A Non-integrable and Non-separable Hamiltonian: \\
Planar Circular Restricted 3-Body Problem (PCR3BP)}
\label{sec:PCR3BP}

PCR3BP is a special case of the gravitational 3-body problem. In addition to a co-planar restriction, it assumes two bodies massive and the third infinitesimal, which models settings like mission design for a space shuttle near Earth and Moon \cite{koon2000heteroclinic}, and understanding a planet's motion around binary stars \cite{li2016uncovering, quarles2020orbital}.
Its Hamiltonian takes the form
\[
\begin{split}
& H\left( \bm{p}, \bm{q} \right) = \frac{p_1^2 + p_2^2}{2} + p_1 q_2 - p_2 q_1 \\
& \qquad - \frac{1-\mu}{\norm{\left(q_1+\mu,q_2\right)}_2} - \frac{\mu}{\norm{\left(q_1+\mu-1, q_2\right)}_2},
\end{split}
\]
with $\mu \in (0, 1)$ a constant mass parameter. Note it cannot be written as $K(\bm p)+V(\bm q)$, hence nonseparable.

In order to focus on comparing with SOTA methods for trajectory accuracy, we predict solutions in the nearly-integrable (non-chaotic) regime of PCR3BP; see Fig.\ref{fig:PCR3BP}. GFNN still has the smallest error among those experimented, and its growth is again linear. SRNN typically performs the best among tested existing approaches, but its published version loses symplecticity in this case due to non-separability\footnote{A possible remedy based on our nonseparable symplectic integrators \cite{tao2016explicit} was mentioned in SRNN as a future direction. This remedy is implemented in a concurrent work \cite{xiong2021nonseparable}, which successfully reduces the error of predicting nonseparable dynamics to the level of SRNN for separable dynamics.\label{ftnt:SRNN_nonseparable}}, and its accuracy deteriorated. Note also that for methods that learn, in the separable case, $V(\bm q)$ in the Hamiltonian or $\nabla V(\bm q)$ in the vector field, now they cannot just do so but have to learn the entire $H(\bm{p}, \bm{q})$ in doubled dimensions.

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
     \includegraphics[width=\linewidth]{cmp_PCR3BP.pdf}
     \vskip -10pt
     \caption{Comparison of PCR3BP orbit predicted by different methods. L-L O-NET (in SRNN paper) is selected instead of SRNN as the Hamiltonian is not separable. Data sequences are prepared with time step $0.1$.
     }\label{fig:PCR3BP}
 \end{center}
\vskip -0.2in
\end{figure}

\subsection{A Discrete-time Non-(Smooth-)Hamiltonian System: the Standard Map}
\label{sec:Results:standardMap}

The standard map is a classical model in accelerator physics. It is a chaotic system whose statistical property is (relatively) well understood. It is a symplectic map in 2D given by
\begin{equation}
\begin{cases}
p_{n+1} &= p_n + K \sin(\theta_n), \\
\theta_{n+1} &= \theta_n + p_{n+1}.
\end{cases}
\label{eq:standardMap}
\end{equation}
The dynamics is genuinely in \emph{discrete} time, as no smooth Hamiltonian ODE can produce a flow map like it\footnote{This is because autonomous Hamiltonian systems in 2D are never chaotic (the Hamiltonian itself is a 1st integral) but the standard map is chaotic.}. $K$ is a positive constant that controls the strength of nonlinearity, and it has been estimated that the region of initial conditions leading to chaos has size increasing with $K$ \cite{chirikov1979universal}.

Methods based on vector fields (e.g., VFNN) or Hamiltonian (e.g., HNN, SRNN) are not very suitable for this prediction task because there is no latent continuous (Hamiltonian) dynamics. One can still apply these methods regardless, for example by using finite differences to construct a fictitious vector field for VFNN and HNN to learn, or just use SRNN without realizing that no Hamiltonian will be able to produce the training data.
Their results (obtained using $h=1$) will be compared with those of GFNN, which is still applicable here as it directly learns evolution maps.

\cref{fig:chaotic_sea_standard_map} illustrates the predicted evolutions of a fixed initial condition in the chaotic sea (of the true dynamics, $K=1.2$) by various methods. Note both $\theta$ and $p$ have been mod $2\pi$ as this quotient compactifies the phase space into the 2-torus without affecting the dynamics (see \cref{eq:standardMap}). Like before, the prediction quality can be inferred from the geometric shape of the set of plotted points, which should match that of the truth (i.e., the latent chaotic sea), and quantitative comparisons can be made using distances between empirical distributions of $p,\theta$ values collected along long time predictions (KL divergences from the truth are provided).

One can see GFNN is the only method that captures the major regular islands (the big holes), but even GFNN does not capture the minor regular islands well. The standard map seems to be a challenging problem; HNN and SRNN did not manage to reproduce any chaotic motion, and VFNN completely distorted the chaotic sea.

\cref{fig:regular_island_standard_map} on the other hand illustrates predictions in regular islands (of the true dynamics, $K=0.6$).
The two (not three, note periodic boundary conditions) elliptical shapes near $p \approx \pi$ and $\theta \approx 0, \pi$ correspond to quasiperiodic orbits, and GFNN is the only one that captures them: the exact trajectory is jumping back and forth between two islands, so does GFNN's prediction, while other methods tend to produce continuous trajectories without capturing the jumps.

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
     \includegraphics[width=\linewidth]{chaotic_standard_map.png}
     \vskip -10pt
     \caption{Predict a chaotic orbit of the standard map. Left 3 columns: the predicted orbit in phase space; right column: marginals of its empirical measure and their KL divergences from the truth.}\label{fig:chaotic_sea_standard_map}
\end{center}
\vskip -0.2in
\end{figure}

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
     \includegraphics[width=\linewidth]{regular_island_standard_map.png}
     \vskip -10pt
     \caption{Predict a regular orbit of the standard map. The 1st plot is orbits predicted by various methods in phase space. The 2nd and 3rd plots shows how their two coordinates change with time.}\label{fig:regular_island_standard_map}
\end{center}
\vskip -0.2in
\end{figure}

\section*{Acknowledgements}
The authors thank the anonymous reviewers who helped significantly improve the quality of this article, and Minshuo Chen for helpful discussions. We are grateful for the partial support by NSF DMS-1847802 and ECCS-1936776.

\bibliographystyle{icml2021}

redpajama_set_name: RedPajamaArXiv


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Not to be confused with A Beam in the Dark.
A Light in the Dark is the secret level of the third world in Yoshi's Island DS. It takes place in a complex castle lit by many torches.
Black Yoshi starts the level near a staircase which the player must go up; then he can continue going right after getting to the top to find more stairs to go down. After that, Yoshi must then jump over a spike pit, collecting Red Coins along the way. Yoshi finds a room inhabited by four Piranha Plants and a flower. A chain is nearby which must then be traversed using Baby DK, he then finds a Middle Ring and then access Area 2.
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At the beginning of Area 4 there is a Middle Ring and a few spikes to traverse, Yoshi has to dodge Crabbles along the way before reaching the goul roulette.
Kurayami ni hio tomose! Light Fires in the Darkness!
The level shares its German name with A Beam in the Dark from Super Mario 3D World.
This page was last edited on April 22, 2019, at 15:38.


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