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Ohio's native residents were killed or driven out by 1800, making the territory safe for civilization and attractive for immigration. At least four Fulkerson branches moved to Ohio from New Jersey and New York between 1810 and 1840. Each branch is linked backed to ancestors on other pages in the Fulkerson Genealogies section. One additional and eventually very sizable branch led by brothers Benjamin and Richard Fulkerson came to Ohio from Virginia, and is the subject of an "unattached branches" page.
New Jersey 1811
Richard P.
Benjamin FULKERSON....bapt. 11 Nov 1753 at the Cranbury Presbyterian Church, Somerset Co., NJ, listed as the son of "Philip Fulkert." He was a grandson of Philip FULKERSON of Somerset Co., NJ and the son of another Philip. [see note] He appeared on the 1778 New Jersey tax rolls, living at Bedminster in Somerset Co. He reportedly died in 1797 at age 44 in Loudoun Co., Virginia. Little else is known about him, except that he was the progenitor of a large Ohio Fulkerson branch. His six known children and many descendants included:
Philip FULKERSON.....b. ca. 1778
Josiah FULKERSON.....ca. 1780-29 Oct 1835, b. in PA per son Gabriel's information in the 1880 Census, m. Margaret GHEEN (b. ca. 1789 in PA), on 13 Aug 1804 in Loudoun, Co., VA, apparently moved to Ohio by 1811.
Amy FULKERSON....b. ca. 1800, m. Henry WARNER on 20 Oct 1825 in Muskingum Co., OH
William FULKERSON....b. 1807 in VA per 1850 Census, he is placed here by process of elimination. By unconfirmed reports, he m. Martha FORD (b. in England ca. 1810) on 31 Dec 1829, probably in Ohio. The 1850 Census showed them (ages 45 and 40) in Dist. 112, Zanesville, Muskingum Co., OH, with seven children ranging from 20 to 5 years. The 1860 Census showed them in Wayne Twp., Muskingum Co., OH, living near Lucy FORD (age 84, probably Martha's mother per another researcher) and near Sarah HARPER (nee Sarah FULKERSON), who would have been William's aunt. Martha's age was again given as 40 on the 1860 Census, and they had one additional child, Edward, born about 1851. William's son John was living with the aunt, apparently assisting her in operating her farm. William and Martha were found again on the 1870 Census, living alone and still farming in Wayne Twp.
Mary J. FULKERSON....b. ca. 1830 in OH per 1850 Census
Louisa FULKERSON....b. ca. 1832 in OH per 1850 Census
William FULKERSON....b. ca. 1836 in OH per 1850 Census
Martha FULKERSON....b. ca. 1839 in OH per 1850 Census
Thomas FULKERSON....b. ca. 1840 in OH per 1850 Census
John FULKERSON....b. Dec 1842 in OH per 1900 Census. He was living at home on the 1850 Census, and living with neighbor Sarah HARPER (nee Sarah FULKERSON), age 77, on the 1860 Census. Sarah would have been his great aunt. John hasn't yet been located on the 1870 Census when he would have been 27. However, the 1880 Census finds a J.H. FULKERSON, age 37 and born in Ohio, living in Dadeville, Tallapoosa Co., Alabama. His father was listed as born in VA. He was living with S.A. FULKERSON, wife, age 45 and born in Ohio. He was last found on the 1900 Census, living at 514 Putnam Avenue in Zanesville, OH, age 57 and widowed, with an 11-year-old son, Charles H. FULKERSON. His occupation was 'drayman.'
Charles Herbert FULKERSON....b. in Zanesville, OH on 9 Nov 1889, as shown by both the 1900 Census and his 1917 draft registration card. Charles was in vaudeville and used the nickname "Rube." His draft card stated he was a 'performer' and the name of his act was "Rube Fulkerson and the Ginger Girls" -- apparently his wife and daughters. He m. Florence (Flo) WAGNER and had at least three daughters who performed on stage with him as the "O'Brien Sisters" -- there was an ad in Variety as late as 1956, announcing a show featuring Mary O'Brien, "daughter of the late Rube Fulkerson, well-known tab comic in days gone by." One son, Charles Jr., died when a month old in 1916. Charles reportedly died in 1948, and Florence in 1962, both in Dallas, TX.
Charles FULKERSON....b. ca. 1845 in OH per 1850 Census
Edward FULKERSON....b. ca. 1851 in OH per 1860 Census
Gabriel C. FULKERSON.....16 Sep 1811- , b. at Salt Creek, Muskingum County, OH, m. Sarah Ann THOMAS (b. 10 Feb 1818 in Muskingum Co., OH) on 3 Mar 1837 at Salt Creek, Muskingum County, OH. The 1880 Census found Gabriel as a 67-yr-old widower and boarder in the household of John A. NEWELL at Salt Creek, Muskingum County, OH.
William Washington FULKERSON....b. in Ohio per his son's death certificate [LDS Familysearch]. He m. (1st) Sarah Catherine JENNINGS on 1 May 1856 and (2nd) Elizabeth Mary ALLEN (b. ca. 1843) on 10 Nov 1866. His three known children were with Elizabeth.
Charles FULKERSON....31 Jan 1867-26 Jul 1921, b. at Chandlerville, OH, was living at 1111 South Johns Avenue when he died at Lima, Allen Co., OH. He was a Methodist minister whose name appeared in Marion Co., OH marriage records about the turn of the century. His wife was Mary. He was buried on 28 Jul 1921 in Toledo, OH. Information on him is from his death certificate [LDS Familysearch].
Ida May FULKERSON....20 Feb 1870, m. Joseph G. TRUEX on 2 Jul 1891 in Muskingum County, OH
Cecilia FULKERSON....b. ca. 1871-
Josiah FULKERSON....Jan 1840-, m. (1st) Sarah C. PLANTS (ca. 1850-1871), m. (2nd) Hannah J. MAWHORTER (Sep 1844-, b. in OH) on 30 Mar 1871 in Muskingum County, OH
Ida FULKERSON....b. ca. 1865
Oliver C. FULKERSON....Oct 1869-
Mary E. FULKERSON....1843-, m. (1st) Townsend HUNT on 19 Sep 1860 and (2nd) Nathan R. PLANTS on 18 Feb 1864 at Zanesville, Muskingum County, OH
Benjamin FULKERSON....1846-, m. Alice BELL
John FULKERSON....b. ca.1871
Blanche FULKERSON....b. ca.1875
Martha A. FULKERSON....1852-, m. Levi J. LEFFLER (8 Oct 1851-, b. at Brush Creek township, Muskingum Co., Ohio). Per a county biographical record, they lived in Bloom township and later in Jerry City, both in Wood Co., Ohio.
Augustus G. FULKERSON....1854-, m. Mary Estelle ERSKINE (b. ca.1858)
Estella FULKERSON....b. ca.1877-
Nisa (?) FULKERSON....b. ca.1879-
William W. FULKERSON....1881-
Warner FULKERSON....17 July 1857-9 Jan 1932, m. Mary Flood GATEWOOD (9 Apr 1860-4 July 1943). The 1880 Census finds them at Salt Creek, Muskingum Co., Ohio, listing Warner as a 23-yr-old farmer. Warner and son Ralph founded a painting business, later known as 'R.L. Fulkerson & Sons Painting,' in Zanesville, Ohio in 1920. The business eventually involved multiple generations of Fulkersons. .
John E. FULKERSON....1879-, m. Maude CROOKS
Della May FULKERSON....1880-, m. Edward VANDEGRIFF
Ella FULKERSON....1880-
Martha Elizabeth FULKERSON....1883-, m. Clarence MASON
Anna Belle FULKERSON....1886-1978
Ralph Laverne FULKERSON....1897-1979, m. Mabel Ada ______ (1903-1986) on 23 Jun 1923, lived at Maysville (close to Zanesville), Ohio.
Benjamin FULKERSON....b. 1816 in VA, d. by 1860, m. Nancy WHARTENBEE (9 Dec 1821-18 Aug 1890, b. in VA) on 12 Apr 1841 in Muskingum Co., Ohio.
Susan FULKERSON....1841-
Alonzo B. FULKERSON....1843-, m. Ella VINCENT Abt 1854 -
Charles B. FULKERSON....1871-
T. J. FULKERSON....b. ca.1877-
Minnie May FULKERSON....b. ca.1880-
William Henry FULKERSON....ca.1848-after 1880, He had at least two marriages, to Juliette WILHITE and Ellen ______ (b. ca. 1857). The order is uncertain at this point. William and Ellen were listed on the 1880 Census, farming at Butler, Darke Co., OH. Only three children - Charles, Mary and Harry - were living with them. [NOTE: Some earlier research appeared to indicate that Ellen was the first wife and that her maiden name was Martha Ellen FULKERSON. If so, she began bearing children at the age of 11. Additionally, her ancestry is uncertain. Further confirmations are needed.]
John FULKERSON....b. ca.1868-, apparently died by 1880
Charles E. FULKERSON....1869-, he's possibly the Charles E. Fulkerson who served as Master of the Ft. Black Masonic Lodge (New Madison), Darke Co., OH from 1898 through 1900.
Sarah Mary FULKERSON....1871-, m. Austin D. DUNN
Benjamin F. FULKERSON....1872-, apparently died by 1880
Harry Epperson FULKERSON....15 Oct 1877-, found in Preble Co., OH when he registered for the draft during WWII (that was required of all men born on or after 28 Apr 1877)
Alice FULKERSON....1879-, apparently died by 1880
Charles E. FULKERSON....1849-
Francis E. FULKERSON....b. ca.1851- [NOTE: The 1900 Census showed a Frank E. FULKERSON with wife Fannie living at Lancaster City, Fairfield Co., Ohio. They had a 2-yr-old son named Frank E. FULKERSON....who later showed up in a Detroit marriage record, marrying Agnes BOTTS on 11 Sep 1919. Connection to the Francis listed here is unknown.]
Thomas FULKERSON....b. ca.1817-, m. Harriet ATKINSON (b. ca.1824). The 1880 Census found them living alone in Wayne, Muskingum Co., Ohio. He was listed as a 63-yr-old farmer, born in OH, with both parents born in VA. Harriet (56) and both her parents were from PA.
Selina FULKERSON....1847-, m. Albert L. MERCER
James C. FULKERSON....1850-, m. Jennie ANDERSON (b. ca.1854)
Howard O. FULKERSON....1872-1965
Susan FULKERSON....m. Marcus S. PORTER on 6 Jun 1841 in Muskingum Co., OH
Selena FULKERSON....b. in in Muskingum Co., OH, m. James N. COCHRAN on 21 May 1848 in Muskingum Co., OH
Alice FULKERSON...nothing known
Margaret FULKERSON....nothing known
Julian FULKERSON...b. ca. 1835, nothing else known
Sarah FULKERSON.....b. ca. 1782. She m. Thomas HARPER on 27 Dec 1803 in Loudoun Co., VA. She was listed on the 1860 Census as Sarah HARPER, 77 and widowed, living a few doors away from an apparent nephew, William FULKERSON, in Wayne Twp., Muskingum Co., VA. Living with her, probably to assist on the farm, was 17-year-old John FULKERSON, son of William.
Benjamin FULKERSON.....ca. 1783-12 Aug 1841, m. Elizabeth (or Alice?) HARPER (1786-1838) on 31 May 1810 in Loudon Co., Virginia. They moved to Ohio by 1820. Information about their descendants is from Chuck Fulkerson's FTM website (see the Cousins E-Mail Page for Chuck's listing and the link to his site).
Mark FULKERSON....b. ca. 1810 in Virginia, his name was listed as Martin on a now-defunct Winstead site which also provided his marriage date. Mark was living alone at the time of the 1830 Census, which found him in Shenandoah Co., VA. He m. Christina WINSTEAD (b. about 1816 in Ohio) on 19 Aug 1841 in Pickaway Co., OH. Mark and Christine appeared on the 1850 and 1870 Census in Pickaway Co., OH. Their known children were:
Hulda FULKERSON....b. ca. 1842 in Ohio
Harvey FULKERSON....b. ca. 1845 in Pickaway Co., OH, m. Louisa M. MERCER on 2 Nov 1865 in Muskingum County, OH
Henry FULKERSON....b. ca. 1849. [The above-noted Winstead site called this child Jimmy.] There was a Henry FULKERSON, b. in VA with both parents born in VA, listed on the 1880 Census at Columbus, Franklin Co., OH. He was a 30-year-old stone mason with a wife Jane, b. in VA ca. 1845. Her parents were both born in VA. Their three children were Samuel (12), Charles (9) and Flora (6) - all born in VA. It's possible this was the same Henry, although only his father was born in VA.
Harriet FULKERSON....b. ca. 1851 in Pickaway County, OH
John FULKERSON....ca. 1853-28 Apr 1903, b. in Pickaway County, OH, d. in St. Joseph County, IN
Ellen FULKERSON....b. ca. 1857 in Pickaway County, OH
Charles W. FULKERSON....May 1859-18 Apr 1918, b. in Pickaway County, OH, d. in Clay Twp, St. Joseph County, IN, m. Lettie BURK on 19 Mar 1880 in St. Joseph County, IN
Lucinda FULKERSON.... 3 Apr 1812-
Sarah Ann FULKERSON....26 Jul 1815-
Adolphus FULKERSON....2 Aug 1817-
Thomas Harper FULKERSON....6 Jul 1820-, b. in Ohio, m. Lydia W. COLLINS (20 Sep 1822-4 Apr 1876, b. Ohio per death certificate of son Alvin) on 14 Nov 1841 in Coshocton County, OH
Josiah Marion FULKERSON....9 Nov 1844- 2 Aug 1924, b. near Plainfield, Coshocton County, OH. In 2007 a cousin forwarded a copy of an article that appeared in The Coshocton Tribune on March 18, 1937. It reported on a book, "History of the Ram Fleet and the Marine Brigade," for which Josiah Marion Fulkerson was listed as the author - albeit as chairman of a committee of brigade surviviors entrusted to write it's history. The article went on to tell Josiah's personal story: he enlisted in Company H, 80th Division, Ohio Regulars, during the winter of 1861-62, soon after turning 17; he fought in the battles of Corinth and Iuka in Mississippi and was wounded at the conclusion of the latter; following his release from convalescence he re-enlisted in the Mississippi Marine Brigade, based at Benton Barracks, Missouri, and served aboard Union gunboats that patrolled and fought along the Mississippi River.
[NOTE: Had he initially joined in this latter endeavor in 1862, he might have fought against one of his cousins, Confederate Captain Isaac D. Fulkerson, in a Mississippi River gunboat battle. As it was, he did fight against one of his cousins, Lt. Colonel Jacob Thompson, at the battle of Corinth. Thompson also went on to maritime pursuits during the war, as a pirate on the Great Lakes.]
The article went on to tell us that, after the war, Josiah "worked his way from the bottom to the top of staff" at a Pittsburgh light manufacturing firm, leaving there after 14 years due to his ill health and moving to Kansas City, MO. Click here to see the article AND a photo of Josiah in the Civil War. From other family research we have determined that Josiah married (1st) Mary Jane COOPER (27 Jul 1839-20 Feb 1881, b. in PA) on 19 Mar 1867 in Allegheny City, Allegheny County, PA. Sometime between late 1876 and December 1878 they moved west to Emporia, Kansas. Mary died in Feb 1881 after bearing 4 children, and Josiah m. (2nd) to Nannie MILLER (11 Feb 1854-11 Jun 1920, thus almost 15 years younger than his first wife) on 29 May 1881 in Kansas City, MO. He must have divorced her between 1893 - when their 7th child was born - and 1896, as he then m. (3rd) Mary A. SULLIVAN (b. 1 Nov 1872, thus 18 years younger than his 2nd wife and more than 32 years younger than his first) on 3 Jun 1896 in the City of St. Louis, MO. About 1917 he opened Fulkerson's Saloon at 6286 Clayton Avenue (at Graham St.) in the Dogtown section of St. Louis. He apparently made at least one enemy in the neighborhood, as he soon came under suspicion of being unpatriotic during wartime. His obituary appeared in the St. Louis Post Dispatch on 3 Aug 1924. Few of his 12 children reached adulthood.
Alpha FULKERSON....6 Dec 1870-15 Dec 1870 at Homewood, Allegheny County, PA
Mary Luella FULKERSON....19 Aug 1872-23 Mar 1873 at Pittsburgh, Allegheny County, PA
Cora Lee FULKERSON....9 Mar 1874-6 Dec 1878, d. at Emporia, Lyon County, KS
Sadie Van Voorhis FULKERSON....26 Sep 1876-, b. at Braddocks, Allegheny County, PA, m. (1st) David J. SCHAMBACH at Hillsboro, Jefferson County, MO on 8 Jun 1904. David died on 15 Sep 1915. 23 years later Sadie m. (2nd) to Harold P. STEEL on 28 Apr 1938 in Montrose, Montrose County, CO
Carrie FULKERSON....b. and d. on 18 Mar 1882 at Kansas City, Jackson County, MO
Frank B. FULKERSON....20 Feb 1883-20 Apr 1883 at Kansas City, Jackson County, MO
Nellie May FULKERSON....6 Apr 1884-3 Mar 1926, m. Edward P. WHEELER on 26 Nov 1903 in the City of St. Louis, MO. Nellie's obituary appeared in the St. Louis Post Dispatch on 5 March 1926.
Ada Charity FULKERSON....31 Oct 1885-, m. Edward C. LOVE on 14 Sep 1906 in St. Charles, St. Charles County, MO
Cora Agnes FULKERSON....21 Jan 1888-Oct 1923, m. a FAGAN. Her obituary appeared in the St. Louis Post Dispatch on 14 Oct 1923.
Mary Alice FULKERSON....11 May 1890-2 Feb 1920, m. Henry W. STAFFORD on 19 Nov 1910 in the City of St. Louis, MO. Henry died a little over 5 years later, on 4 Mar 1916, and Mary Alice died less than 4 years later at the young age of 29. Her obituary appeared in the St. Louis Post Dispatch on 3 Feb 1920.
Marion FULKERSON....5 Feb 1893-
Sylvester Verne FULKERSON....17 Jul 1898-13 Jul 1910. His obituary appeared in the St. Louis Post Dispatch on 14 Jul 1910.
Charity Ellen FULKERSON....23 May 1847-9 Dec 1921, b. in Coshocton County, OH, d. at Columbus, Franklin County, OH. She married as his second wife Charles B. JOHNSON (b. 30 Apr 1840 in Tompkins County, NY, son of Henry L. and Ocee Ann (BROWN) JOHNSON) on 7 Jul 1867 in Coshocton County, OH; his first marriage was to Martha BALTZALL, daughter of Joseph and Lydia, from 7 Mar 1863 to 12 Jul 1865. Thus Charles had two mothers-in-law named Lydia. A Civil War veteran (24th Ohio Volunteer Infantry, 1861-1863), Charles was in the harness-making trade. In 1874 he patented a diamond trace buckle, for which he received about $1,000 - probably equal to a year's income in most trades of that day. He set up his 3rd and final shop in New Castle Township a few months after marrying Charity, and there they spent the remainder of their lives. He served three terms as Justice of the Peace, and they were proprietors of the Union Hotel, which was described as doing a fair business. Three of their four children (one had died young) were listed in 1881:
SOURCE: HISTORY OF COSHOCTON COUNTY, OHIO: ITS PAST AND PRESENT, 1740-1881, AUTHOR: N.N. HILL, JR.
Blanche JOHNSON....b. 6 Jul 1868
George Washington JOHNSON....b. 10 Oct 1869-May 1936, d. in Franklin Co., OH and buried at Greenlawn Cemetery there. He m. Katherine GIFFEN, dau. of Warren GIFFEN and Harriet WILLIAMS.
Dema Reay JOHNSON...19 Jul 1890-27 Jan 1969, b. Kansas City, KS, d. in Franklin Co., OH. She m. Frank B. OSBUN (1885-1926) by 1909. They had two daughters and two sons, per a BUTTERMAN descendant.
Robert JOHNSON....b. 21 Feb 1877
Sarah Alice FULKERSON....23 Jun 1851-28 Jan 1936, b. in Coshocton County, OH, m. Esastus Newton POWELSON (1844-1918) on 3 Jan 1871 in Coshocton County, OH
Amanda Jane FULKERSON....29 Jun 1854-18 Aug 1893, b. in Coshocton County, OH, d. at Kansas City, Jackson County, MO
Alvin C. W. FULKERSON....10 Jul 1858-3 Apr 1911, b. in Coshocton County, OH, d. at Kansas City, Jackson County, MO. He was a manager at Phoenix Bottling at the time of his death, married and living at 2324 Olive Street in Kansas City. The informant on his death certificate was Elmer C. Fulkerson. (He may have been the Fulkerson mentioned in the New Oxford Item of New Oxford, PA on 19 Jun 1891: "Over 1000 bits of skin have been successfully grafted upon the legs of A. C. Fulkerson, of Kansas City, who had stepped in boiling grease.")
Ella Clayborn FULKERSON....5 Jan 1864-27 Dec 1903, d. at Canton, Stark County, OH
Charles Sherman FULKERSON Sr.....27 Dec 1865-11 Oct 1948, b. in Linton Co., OH, d. at Kansas City, Jackson Co., MO, where he lived for 61 years. He m. Alice Marguerite RAFTISY (1868-1930) on 21 Jun 1893 in Kansas City, Jackson County, MO. She preceded him in death.
Ella Charity FULKERSON....28 Mar 1895-5 Jun 1984, b. in City of St. Louis, MO, d. in Kansas City, Jackson County, MO
Charles Sherman FULKERSON, Jr....26 Apr 1897-Feb 1975, in Kansas City, Jackson County, MO. He was the informant for his father's 1948 death certificate. His father was living with him at 6012 Charlotte Street at that time.
Margaret FULKERSON....22 Jul 1822-, b. in Ohio
William FULKERSON.....b. ca. 1790, he is still being researched. William apparently remained in Virginia while his siblings emigrated to Ohio. He married an Amelia by 1818, per a Loudoun County land record. He was on the 1830 Census for Loudoun Co., VA, apparently widowed with two young boys.
Anne FULKERSON.....b. ca. 1792, m. William BLOCKSOM (b. Milton, Sussex Co., DE on 18 Sept 1785) on 5 Jun 1818 (date also reported as 30 Aug 1820) in Muskingum Co., OH. One of their great-great-grandsons was the macabre author/illustrator, Edward GOREY (1925-2000). He created the animated introduction to the PBS series "Mystery!" and helped create the 1977 Broadway production of "Dracula" - which earned him a Tony Award for Best Costume Design and a nomination for Best Scenic Design.
Joseph FULKERSON....7 Mar 1755-11 Jul 1840, bpt. 20 Apr 1755 at Raritan, NJ. Joseph was a grandson of Joseph VOLKERTS and son of Philip FULKERSON and Maergrite/Margrita FARLEY. He enlisted in
Patriots in Prison Ships
Conveyed to York we found, at length, too late,
That Death was better than the prisoner's fate
There doomed to famine, shackles, and despair,
Condemned to breathe a foul, infected air,
In sickly hulks, devoted while we lay,--
Successive funerals gloomed each dismal day
The various horrors of these hulks to tell--
These prison ships where Pain and Penance dwell,
Where Death in ten-fold vengeance holds his reign,
And injured ghosts, yet unavenged, complain:
This be my task--ungenerous Britons, you
Conspire to murder whom you can't subdue
by Philip Freneau, captured at sea on the
Aurora in 1780 and made a prisoner on the
prison ship Scorpion in New York harbor.
the fall of 1775 to serve under Capt. John Polhemus in Col. William Wind's 1st Jersey Regiment, completing this service at Fort Ticonderoga, NY in 1776. In 1777 he enlisted for 3 years under Capt. Andrew McMyers in Col. Ogden's regiment of the New Jersey Line (Continental Army) and participated in the battles at Brandywine and Germantown. At Germantown he was wounded and Capt. McMyers was killed. The name of Joseph Fulkerson appears on the muster rolls at Valley Forge, during the winter of Washington's famous encampment there, from December 1777 through May 1778, listed as a Private in Van Angel's company of the New Jersey Brigade, 1st New Jersey Regiment. He rose to the rank of sergeant and participated in the Battle of Monmouth in June 1778. An August 1778 payroll found in his pension file showed that he made 3 pounds, 15 shillings per month as a sergeant.
Joseph was discharged by Lt. Col. Francis Barber at Mendam Huts near Morristown, NJ in January 1780, and not long afterward joined the crew of an American privateer (a 'legalized' pirate ship) hired to capture British merchant vessels. Their luck ran backward, and they were themselves taken captive. Joseph spent over a year on a British prison ship in New York harbor, where he "suffered much from cold, hunger and sickness" per his later pension claim. [His name does not appear on the prisoner roster of the most infamous British prison ship - the "Jersey" - at http://www.usmm.net/revdead.html, so it is likely he was on one of the many other prison ships there.] Joseph was paroled or exchanged in the fall of 1781, about the time of Corwallis' defeat at Yorktown. In 1818 he applied for and received Revolutionary War Pension #S42307.
In 1789, according to GODDING family research, he m. Betsy OLIPHANT (30 Oct 1764-15 Jul 1840, b. at Raritan, NJ, dau. of Andrew Oliphant Jr, son of a Scotch immigrant) of Morristown, NJ, at the 1st Reformed Church in Raritan. He was listed as Joseph FALKERSON on the 1810 Census at Plainfield, Northampton Co., PA, age over 45, with wife and 3 children. By December 1838 they moved to Monroe Co., Ohio - probably because some of their children were moving there - and he applied to have his pension transferred to Ohio. They appeared on the 1840 census, but both died in July of that year, just four days apart. Both were buried at the Pioneer Cemetery in Woodfield, OH. In the following month, the Monroe County Court of Common Pleas, [Journal 4, Pg 2 -- August 1840 Term, 6th Day] recorded that:
"It was proven to the satisfaction of the court that Joseph Fulkerson, late a Revolutionary pensioner of the United States died in Monroe County on the 11th day of July 1840 leaving a widow who survived him but 4 days and died on July 15, 1840, leaving Joseph Fulkerson, Sarah Ann Fulkerson, Thomas Fulkerson, and Margaret L Fulkerson, their children and heirs."
Joseph Farley FULKERSON....1796-11 Apr 1874, b. in NJ, lived in Northampton Co., PA by 1810 Census. Joseph's middle name was called "Farling" in a son's 'vanity biography' of a century later. He called himself 'Joseph F. FULKERSON.' It is very likely the 'F' actually stood for FARLEY, his grandmother's maiden name.
Joseph m. Margaret MESSINGER (13 Oct 1801-17 Apr 1871, daughter of Philip MESSINGER and Margaret PILE), probably about 1819 in Easton, PA. Joseph was appointed as a guardian of Philip MESSINGER's younger children, as evidenced in the Journal of the Pennsylvania Senate on 5 Dec 1820 (click here to see this document). Joseph witnessed papers related to the will of Margaret's grandfather, John A. PILE, in Easton in 1823. He moved to New York City to engage in manufacturing about 1826 and later went upstate to Steuben Co., NY, per son George's biography. They journeyed west to Ohio about 1838 — apparently accompanied by Joseph's parents, brothers and at least one married sister, per his father's pension file. On the 1840 Monroe Co., OH census they were living at Woodsfield and had 3 boys and 4 girls. In 1846 they put their land up for sale and moved to Bucyrus, Crawford Co., OH. A Linn County, Iowa biography, written of his son William in 1901, tells us that
In 1856 he came with the family to Linn county, Iowa, the journey being made by teams, and they camped and cooked by the roadside during the four weeks en route. They first located on what is known as the Biggs' farm, between Cedar Rapids and Marion, but after residing there for six months they removed to a farm near Iowa City in Johnson county. The winter of 1856-7 being unusually severe, the crops were almost a total failure, and Mr. Fulkerson, like many others became so discouraged that he returned to his old home in the Buckeye state, the journey being made by way of the Mississippi and Ohio rivers. Only three of the children had removed with the parents to Iowa, though two daughters had previously located there.
His son George's biography omits the Iowa adventure and tells us simply that Joseph Farley FULKERSON, "subsequently moved to Bucyrus, Crawford County, OH, where his death occured April 8, 1874. He was followed to the better land that same year by his wife, who, like himself, was a consistent member of the Methodist Episcopal Church." They are buried at Oakwood Cemetery, Crawford Co, OH.
Rachel FULKERSON....b. ca. 1820, probably at Easton, PA. She was still living at home at the time of the 1840 Census, listed as a female aged 20-29. She married, as his second wife, Kinney A. POWELL (1817-after 1899) at Bucyrus, OH on 24 Mar 1856. The 1860 Census found them in Scott Township, Johnson Co., Iowa (her age was listed as 40). Per an 1854 property grant for "Kinney A Powell" of Bucyrus, Crawford, Ohio, they had land in Iowa City, Johnson Co., Iowa. The 1880 Census showed them at Iowa City with two sons born from their union. The 1900 Census found him at Des Moines, widowed, living with two children. He reported his occupation as "capitalist."
Joseph K. POWELL.....ca. 1857-, b. in Iowa, was 23 and employed at a cutlery works in 1880. He married Adalaide PATTERSON (b. ca. 1859 in OH) in Blissfield, Lenawee Co., Michigan on 11 Jan 1883.
John F. POWELL.....b. ca. 1859 in Iowa, was 21 and not working at the 1880 Census. The 1900 Census stated he was b. in July 1857, was still single, working as a painter and carpenter, and living with his father.
Tobias FULKERSON.....1822-1873, b. in PA, m. Mary Ann WINDLAND about 1848 in Monroe Co., OH, and possibly lived there until 1865 when he moved near Gatchel, Perry Co., Indiana. Tobias died 8 years later, and his wife 5 years after him. Their known children:
Margaret E. FULKERSON....1849-1871, b. in Ohio, m. William CRONIN (b. 1848 in Monroe Co., OH) on 8 Nov 1870 in Perry Co., OH. William m. (2nd) on 10 Feb 1872, to Elizabeth E Kincaid (b. 1852) in Perry Co. Margaret is buried at Slaughter Cemetery, Perry Co.
William Alexander FULKERSON....6 Jan 1851-3 Apr 1912, b. in Ohio, m. Margaret Susan DOUGHTY (b. 30 Apr 1850, daughter of John DOUGHTY and Celia/Selina WAIT) by 1877. They apparently moved to Arkansas by 1894. Both died at Center Ridge, Conway Co., Arkansas and are buried at Center Ridge Cemetery.
Mary E. FULKERSON....1877-, b. in Indiana
Gilbert T. FULKERSON....1879-13 Nov 1942, born and died in Perry Co., Indiana. In the 1900 Census he was listed in the household of his uncle, Albert Seymour Fulkerson. He m. Mary Eliza CRONIN (daughter of Sylvester CRONIN and Mary PANNETT) on 30 Nov 1902 in Perry Co., Indiana. She was born in 1878 and died 9 Jul 1939 -- like her husband -- in Perry Co. They had nine children, found on the 1920 and 1930 Census. On the latter Gilbert was listed as Bert. Their final home was at Tell City, Perry Co., Indiana. They and several of their children are buried at the Lilly Dale Cemetery in Gatchel. [Note: LILY DALE, a 1985 song by Natalie Merchant, was apparently named for this cemetery]
Lizzie FULKERSON...age either 16 or 18 on 1920 Census
Minnie FULKERSON...age 14 on 1920 Census
William Sylvester FULKERSON....age 12 on 1920 Census, b. 2 Jun 1907 per a descendant.
Albert A. FULKERSON....26 Oct 1910-6 May 1988. Listed on the 1920 and 1930 Census as "Addison FULKERSON"
James Edward FULKERSON....6 Oct 1911-Mar 1974, m. Mary ___, had two children.
Bertha M. FULKERSON....13 Mar 1914-21 Nov 1967.
Charles Frederick FULKERSON....16 Jul 1916-20 Feb 1963, 3 children.
Porter FULKERSON
Myrtle Kate FULKERSON....17 Oct 1881-1970, b. in Perry Co., IN and known as Kate, m. Walter Roscoe CRONIN on 7 Oct 1900 in Perry Co., IN
Eva L. FULKERSON....born May 1886 in Perry County, Indiana
Lessel J. FULKERSON....born Nov 1888 in Perry County, Indiana
Rosa A. FULKERSON....born June 1894 in Arkansas
Jacob Theodore FULKERSON....1854-, b. in Ohio, may be buried at Center Ridge Cemetery with no marker, beside brother William
James Hardy FULKERSON...1856-13 Nov 1914, b. in Ohio, moved to Center Ridge, Conway Co., Arkansas. He m. (1st) Sarah F. GREER on 9 Sep 1883 in Conway Co., Arkansas, then m. (2nd) Margaret ATKINSON (born in Arkansas in Aug 1868, daughter of Joseph ATKINSON and Nancy MAHAN) on 11 Sep 1887 at Center Ridge, Conway Co., Arkansas. They were reported at that place in the 1900 Census. He was buried at Center Ridge Cemetery, Conway Co., Arkansas.
Mandy FULKERSON....was not in her parents' household in 1900, m. Perry GRAHAM
Mary M. FULKERSON...b. Jun 1888, listed on the 1900 Census.
William Allen FULKERSON...7 Aug 1891-4 Jun 1954, born in Arkansas. Listed as Allen on the 1900 Census. He married Mary Elizabeth (Lizzie) MARLOW (27 Nov 1886-8 Mar 1950, born in Missouri) and had five children. They lived and are buried together at Center Ridge, Conway Co., Arkansas. Their headstone lists him as Allen Fulkerson.
Scott Audry FULKERSON...ca. 1908-
Edgar Jay FULKERSON...ca. 1913-
Zula Mae FULKERSON...9 Feb 1915-
Alden FULKERSON...known as Tom, served in WWII, lived at Morrilton, AR
Audell FULKERSON...
Nora E. FULKERSON...b. Jul 1893, listed on the 1900 Census. She m. Homer HARRIS
Nancy Beulah FULKERSON...b. ca. 1904
George W. FULKERSON...1858-
Albert Seymour FULKERSON....20 Nov 1869-6 Apr 1938, b. in Perry County, IN, m. Sidney WILLIAMS (1875-May 1899, daughter of Henry and Nancy WILLIAMS) on 28 Jan 1894 in Conway Co., Arkansas. Following her death (related to the birth of their fourth child) he married her older sister Margarette (b. Aug 1869) on 7 Apr 1900 and had five more children. Albert and Margarette may be buried in Grandview Cemetery with no marker (in Conway Co., AR, not far from Center Ridge).
Benjamin Franklin FULKERSON...Apr 1896-
Lydia (Liddie) FULKERSON...Dec 1897-
James Garfield FULKERSON...31 Aug 1898-8 Mar 1952, buried at Pleasant Hill Cemetery, Conway Co., AR
Lawson M. FULKERSON...20 May 1899-23 Mar 1975, lived at Springfield, Conway Co., Arkansas, m. Lula ___, had five children, is buried at Pleasant Hill Cemetery.
Corene FULKERSON...
Roosevelt FULKERSON...b. ca. 1903
Dorsey L. FULKERSON...19 Apr 1904-28 Dec 1979, lived at Cabot, Lonoke Co., Arkansas, buried at Little Rock National Cemetery
Manuel R. FULKERSON...b. ca. 1906
Roy FULKERSON...b. ca. 1917
Margaret Ann FULKERSON....17 Nov 1823-1 Dec 1901, m. David Charles MORRIS (26 Apr 1823-21 Jan 1916) on 4 May 1843 in Monroe Co., OH. He was the son of Henry Gustler (Gusler) MORRIS (b.3 Dec 1798; d.6 Nov 1875) and Mary KINKADE (b. 1 Feb 1804; d. 9 Oct 1865, per a Bible record cited at a now-defunct MORRIS family site. Margaret and David moved their family of 10 children to Perry Co., Indiana in 1863 by boat on the Ohio River, and spent the remainder of their lives there. David died at Ranger, Perry Co., IN.
Joseph Henry MORRIS....17 Mar 1844-12 Mar 1899, b. in Pennsylvania
John Atkin MORRIS....1 Dec 1845-31 Jul 1846
Thomas Webster MORRIS....1 Nov 1847, Monroe, OH; d. 11 Dec 1929, Lincoln City, Spencer, IN. He married SUSANNA ROWLEY 13 Jun 1870 in Perry Co., IN
Sarah E. MORRIS....b. Abt. 1871 in Indiana.
Alexander J. MORRIS....b. Abt. 1874 in Indiana.
Charles MORRIS....b. Abt. 1877 in Indiana.
James Buchanan MORRIS....27 Dec 1879-1 Aug 1938, b. at Lincoln City, Spencer Co., IN; d. at Yuma, AZ. He married Susie D. ROBINETT on 21 Apr 1904 in Malden, Dunklin Co., MO
Mabel Mary MORRIS....27 Mar 1905-25 Jul 1959, b. at Malden, Dunklin Co., MO, d. at Bellflower, Los Angeles Co., CA. She married DANIEL _____ on 25 Apr 1925 in Piggot, Clay Co., AR.
Mary E. MORRIS....b. Abt. 1889, Indiana.
George Sylvester MORRIS....11 Nov 1849-21 Mar 1923, b. in Monroe Co., Ohio, d. at Owensboro, Daviess Co., KY. He married Elizabeth Hannah PANNETT. They had 11 children, per a descendant who named only one of them:
Sarah Ann MORRIS....married John Walter ROWLEY in Tell City, Perry Co., Indiana.
William Frank MORRIS....22 Feb 1852-1937.
Rachel Jane MORRIS....31 Mar 1854-31 May 1953, b. in OH
Mary Henrietta MORRIS....25 Feb 1856-10 Nov 1923, b. in Monroe Co., Ohio, d. in Warren Co., Kentucky
David Louis MORRIS....13 Sep 1858-5 Apr 1885, d. at Perry, IN.
Tobias T. MORRIS....28 Aug 1861
John W. MORRIS....24 Feb 1864-12 Oct 1930
Dr. George W. FULKERSON....1 Dec 1828-28 Dec 1892, b. in New York City and educated in Steuben Co., NY, d. at Houston, Shelby Co., OH. George signed a deed for Joseph Farley Fulkerson in Monroe Co., OH, presumably after Joseph moved to Bucyrus. He m. (1) Elizabeth NISHWITZ (d. 1858) on 1 Jan 1857 in Miami Co., OH [county marriage index] and (2) Margaret MILLS on 29 Jun 1876 in Shelby Co, OH. George served the community during the Civil War: he provided free medical care to every family that sent a soldier off to war, according to his biography. He lived in Cynthian Twp, Shelby Co, OH when he wrote his will on 7 Apr 1891, naming his wife and 3 sons. He was buried in that county at Houston Cemetery, Section 1, Lot 9.
John Medaris FULKERSON...3 Apr 1878-, m. Maude L. KILLIAM. His biography appeared in the 1920 Memoirs of Miami Valley, Ohio:
John M. Fulkerson. A prosperous and growing commercial enterprise at Troy is found in the up-to-date and well-equipped pharmacy of John M. Fulkerson, who has been proprietor of this business for the past eighteen years. Mr. Fulkerson is a native of Shelby county, Ohio, born April 3, 1878, a son of George W. and Margaret J. (Mills) Fulkerson, whose children were: John M.; Walter, identified with the steel industry at Canton, Ohio; and William a lieutenant in the United States Navy. The father of Mr. Fulkerson was a physician who practiced for many years in Shelby County and the youth was brought up amid surroundings that naturally inclined him toward his present calling. After attending the local schools, he took up the study of pharmacy at Ohio Northern University, and after his graduation from that institution began to follow the drug business. In 1902 he came to Troy, where he established himself in business, and since that time has built up a large and satisfying patronage among the leading people of this community. He has a well-equipped establishment and carries a full line of drugs, medicines, toilet articles, cigars, candies, etc., and the manner in which he has directed his transactions has given him business standing and public confidence. Mr. Fulkerson is a member of the Masonic fraternity. He has always given his unquestioning support to movements of a worthy civic character, and during the period of the war was prompt in his response to requests for contributions to the various movements promoted for the success of American arms. Mr. Fulkerson married Maude L., daughter of William Killiam, of Piqua, and to this union there have been born two daughters: Margaret, who is her father's able assistant in the drug store; and Florine, born in 1909, who is attending the graded schools of Troy.
Walter Roy FULKERSON...in 1920 was associated with the steel industry at Canton, Ohio
William A. FULKERSON...in 1920 was a lieutenant in the US Navy
Elizabeth FULKERSON....9 Apr 1829-22 Aug 1880, m. on 27 Aug 1850 to William PATTERSON at Bucyrus, OH, d. in Linn Co., Iowa.
Eliza Jane FULKERSON....12 Mar 1831-18 Dec 1910, m. Robert PATTERSON on 30 Nov 1852 at Marion, Marion Co., OH, d. at Byran, Williams Co., OH
James Wesley FULKERSON....1839-Apr 1881, b. in Monroe Co., OH, m. Isabelle YOUNGMAN on 22 Dec 1863 at Bucyrus, OH and died there just 18 years later. This is probably the James who served as a Lieutenant in the 8th and later the 136th Ohio Volunteer infantries during the Civil War. The 8th was a short-lived outfit: it "reported at Camp Chase May 12, 1861, was mustered, uniformed, armed, equipped, and shipped to Washington City. It remained on garrison duty in and around Washington during its entire term of service, which expired August 20, 1861, when it was mustered out and discharged."
Joseph Theodore FULKERSON....24 Jan 1843-5 Jul 1913, b. in Monroe Co., OH, m. Mary Ann CURLY on 14 Aug 1872 at Athens, OH, d. at Columbus, OH
William Alexander FULKERSON....17 Oct 1847-3 Apr 1912, b. at Bucyrus, OH. m. Maggie YANT on 4 May 1869 at Columbus, OH, d. at Bucyrus, OH. At age 9 his family moved to Iowa, but soon returned to Ohio. There he attended school until he was 16, and then joined the Union army in 1864. He spent his 17th birthday participating in his 6th battle. In early 1865 his unit was surprised by Confederates and captured, but William and about 100 other enlisted men escaped and made it back to Union lines. In 1866 he developed an interest in building, which led to a move to Chicago in 1871 to train in architecture. The Linn County biography of him, from 1901, then tells us that he moved to Cedar Rapids, Iowa, where he
built the Soldiers Home at Marshalltown; the Central Hall building of the State Normal School at Cedar Falls in 1895; the Auditorium of Cedar Rapids, which has a seating capacity of thirty-five hundred, in 1899; and ten of the public school buildings of this place, besides numerous fine business blocks and residences.
Huldah E. FULKERSON....m. John GODDING; the Monroe County, Ohio, Court of Common Pleas did not list her as a child of Joseph FULKERSON in 1840. In 1858, according to Joseph's pension file, Emily R. GODDING of Calumet County, Wisconsin, applied for 160 acres of bounty land in that county. She made this claim as an heir of Joseph FULKERSON, in accordance with an Act passed by Congress on 3 Mar 1855, and stated she was under 21 years of age on the date of that Act. On 11 March 1859 Huldah filed a claim for survivor benefits.
Joseph Oliphant GODDING....b. 22 Feb 1821 at Reading, Steuben Co., NY
Morton Duane GODDING....
Sally Ann GODDING....b. 1 Apr 1823 at Reading, Steuben Co., NY
Tallmai Farrell GODDING....b. 10 Jun 1825 at Reading, Steuben Co., NY [His name comes to us from a Godding source -- could it be Tallman?]
Jared Colbert GODDING....b. 8 Oct 1827 at Reading, Steuben Co., NY
James Henry GODDING....b. 29 Jul 1830 at Reading, Steuben Co., NY
John Theodore GODDING....b. 11 Mar 1832 at Greenwood, Steuben Co., NY
Asa Robins GODDING....b. 10 May 1834 at Greenwood, Steuben Co., NY
Margaret Albina GODDING....b. 19 Nov 1838 at Greenwood, Steuben Co., NY
Emily Rosemond GODDING....b. 27 Oct 1842 in Allegheny Co., NY
Sarah Ann FULKERSON....m. Jarred THOMAS; late in life she went to live with a daughter in Kansas
John FULKERSON....On the 1820 census of Steuben Co., NY, he is shown as a neighbor of his uncle Caleb FULKERSON, living at what would become the town of Starkey in Yates Co. A family legend says that one of Joseph's sons drowned in a lake in New York in the 1820's. It may have been this John.
Margaret Letitia FULKERSON....m. Henry JOHNSON (b. 1777) as his 2nd wife, probably about 1840 when he was at least 62 years old. Henry served in the War of 1812 and was a minister and mayor of Woodsfield, Monroe Co., Ohio. In 1850, according to her father's pension file, Henry JOHNSON stated his wife was an heir of Joseph FULKERSON and requested survivor benefits, calling himself a "poor old feeble man Destitute of many of the comforts of life."
Thomas FULKERSON.....nothing is known about him, except that he was alive in 1840 when the Monroe County Court of Common Pleas recognized Thomas as a child of Joseph FULKERSON the Revolutionary War veteran
John FULKERSON....b. 1780 in Steuben County, NY, d. in Huron County, OH by 1837, was a grandson of Philip FULKERSON of New Jersey and son of Volkert FULKERSON. Based on his age and location, he may have been the John Fulkerson who served as a fifer in Colonel Henry Bloom's Regiment (Tompkins County), New York Militia, in the War of 1812. [Index to the Compiled Military Service Records for the Volunteer Soldiers Who Served During the War of 1812, M602, National Archives and Records Administration] He m. Sarah (Sally) LEWIS (b. September 06, 1796 in Seneca, NY) in Cayuga (Tompkins) Co. in 1813. They moved to Catharine, NY before 1820. Sometime after 1820 they moved to New Haven, Huron Co., Ohio. Their first 4 children were born in NY, and the second 4 in Ohio. After John's death, Sarah LEWIS m. (2nd) Isaac BOWERS in 1837 and (3rd) Alfred MOFFIT in 1840, both in Huron Co., Ohio.
Cornelia FULKERSON....m. October 23, 1834 to ________
John FULKERSON
Jasper FULKERSON......m. Clarissa TRUCKS (b. 11 Feb 1819 in Richland Co., OH, d. 1 Jun 1889 in Williams Co., OH; per a TRUCKS family site, she was the dau. of William TRUCKS of PA [?] and Lydia FULLER of CT) on 30 Nov 1835 in Richland Co., OH, lived in that county at Mansfield, later moved to Williams Co., OH. BLM patent records (document no. 11478) shows he purchased 40 acres in Darke Co., OH on 1 Jun 1845. They had 2 known children. It is possible they divorced by 1844, as the TRUCKS site reports Clarissa m. (2nd) to Peter Stryker VAN NORTWICK (1787-1854) on 20 Dec 1844 in Williams Co., OH, and (3rd) to Sam SHAFFER (b. 1815) on 1 Sep 1855 in Williams Co., OH.
Sarah Ann FULKERSON....b. 21 Jul 1839, m. William Young FICKLE in 1855 at Williams Co., OH. She had four children, all of them quite Fickle:
William Irving Morse FICKLE
Evelyn A. FICKLE
Alvery Wooster Baker FICKLE
Nancy Clarrissa FICKLE
John Trucks FULKERSON....18 Oct 1840-26 Nov 1924. Per a descendant who wrote in 2001, born in Ohio, died in Yakima, Yakima Co., Washington. This was echoed on a McGAHA family site, which reported his wife and children. John m. Ellen Elizabeth HIATT (b. 2 Oct 1851 at Potawatomy, Cass Co., IA, d. 13 Sep 1936.) Her HIATT line traces back to George HIATT, born 1698 in Glastonbury, Somerset, England, who emigrated to Bucks Co., PA, before 1730. They had nine children:
Clara Ann FULKERSON....b. 27 Jun 1867 in Ashland, Jackson Co., OR
George Edward FULKERSON....b. 7 Dec 1868 in Keno, Klamath Co., OR
Sadie Mae FULKERSON....b. 24 Aug 1871 in Keno, Klamath Co., OR
Calvin Thomas FULKERSON....b. 21 Sep 1872 in Keno, Klamath Co., OR, m. Susan A. HAYES (b. Dec 1874 in MO) on 28 Feb 1892 in Naches, Yakima Co., WA
Joyce FULKERSON....
Valney E. FULKERSON....b. Feb 1893 in Washington
Della Ray FULKERSON....b. Aug 1897 in Yakima, Yakima Co., Washington
Marsha Bell FULKERSON....b. 16 Mar 1875 in Langell Valley, Klamath Co., OR
Lewis Hiatt FULKERSON....b. 2 Mar 1877 in Langell Valley, Klamath Co., OR, middle name also spelled Hyatt, died 28 Dec 1924 in Grandview, Yakima Co., WA. He was married to Celia FERGUSON.
Nettie Rae FULKERSON....b. 19 Nov 1878 in Langell Valley, Klamath Co., OR
Jasper Thurston FULKERSON....b. 27 Oct 1882 in Langell Valley, Klamath Co., OR
Lenna Evalin FULKERSON....b. 25 May 1888 in Langell Valley, Klamath Co., OR
Harriet FULKERSON....b. September 08, 1820, m. 3 Oct 1839 to ______
Liza FULKERSON....b. aft. 1820 in Huron County, OH
Maria FULKERSON....b. aft. 1820 in Huron County, OH
Lewis FULKERSON....11 Aug 1826-1898, b. in Huron County, OH, m. Mary Ann STRINGER (4 Feb 1829-22 Feb 1896, b. in Richland Co., OH, daughter of John STRINGER [1799-1871] and Esther JOHNSON [1799-1865]) on 12 Aug 1847 in St. Joseph Co., IN. Mary Ann died in Clay Twp, St. Joseph Co., IN and was buried at Tutt Cemetery. [NOTE: Darke County, Ohio marriage records show that a Lewis FULKERSON m. a Miriam DOWLER on 10 March 1864. Possibly the same Lewis?]
Sarah J FULKERSON.....b. 29 June 1848 South Bend, St. Joe Co., IN, per information she apparently wrote in the ledger at one of the 'Old Settlers Reunions' held each year in Greene Co., Iowa during the late 1800s. The entry indicated her father's birthdate was 11 Aug 1826 and her mother's 4 Feb 1829. Sarah married ____ HELLER and resided in Jefferson, Iowa.
John FULKERSON......10 Jul 1850-28 Apr 1865, buried at Tutt Cemetery
Frank L FULKERSON......26 Oct 1859-, b. in St. Joseph Co., IN, m. Ida A. WEAVER, resided in Niles, MI. "A Twentieth Century History of Berrien Co, Michigan," Chapter XXVI, includes a sketch on Frank L. Fulkerson, describing his parents as early settlers of St Joseph Co., IN.
Oren Adam FULKERSON...10 Dec 1883-, lived in South Bend, Indiana and near Edwardsburg, Cass Co., Michigan
Charles FULKERSON.....the 1895 "Standard Atlas of St. Joseph County Indiana" lists a 'C.W. and A. Fulkerson' in Clay Twp. Father Lewis is listed as having 2 properties nearby.
Orsen FULKERSON.....4 Jun 1864-28 Sep 1865, buried at Tutt Cemetery
Edwin (Edward?) L FULKERSON.....1866-1915 buried at Tutt Cemetery
daughter FULKERSON.....married Frank JENNINGS, resided in Elkhart Co., IN
daughter FULKERSON.....married John Q. JOHNSON, resided in Clay Twp.., St. Joseph Co., IN
daughter FULKERSON.....married Will McCOMBS, resided in Clay Twp., St. Joseph Co., IN
Sophia FULKERSON....b. aft. 1820 in Huron County, OH
Richard P. FULKERSON....b. ca. 1803, was a grandson of Philip FULKERSEN and son of Dirck FOLKERSON of Brunswick, Middlesex Co., NJ. Richard m. Sarah Ann WYCKOFF (22 Jan 1807-1874) in Somerset Co., NJ on 29 Oct 1829. They apparently moved to Ohio by 1833 and probably settled in Ashland County. They had at least 8 children - a 9th, Jeremiah, is added here based on dates, proximity, family naming patterns and Ashland Co. marriage records:
Mary FULKERSON....b. 18 Aug 1830
Jeremiah FULKERSON....m. Mary A. SWINEFORD on 26 Nov 1863 in Ashland Co., OH. NOTE: A photographer named J.T. FULKERSON advertised daguerreotypes and ambrotypes in Ashland, Ohio, in 1859-1860.
John B. FULKERSON....17 Sep 1833 in OH, m. Carrie E. GADDORD (b. ca. 1838, NY) on 13 Mar 1865 in Ashland Co., OH, moved to Cleveland shortly thereafter
Frank W. FULKERSON.....b. ca. 1867 at Cleveland, OH
Fannie FULKERSON....b. ca. 1873 at Cleveland, OH
Elias FULKERSON....
Emma L. FULKERSON....b. 19 Jun 1839 in OH, m. Henry J. CRESSINGER on 30 Jun 1863 in Ashland Co., OH
Harriet Poe CRESSINGER ....b. ca. 1872 in Ashland, Ashland Co., OH, m. Herbert Rodman PALMER on 9 Dec 1896 in Ashland, Ashland Co., OH per a CHESEBRO family site.
Jane FULKERSON....b. 27 Feb 1842 in Ohio
Loren FULKERSON.....b. 15 Oct 1844 in Ohio
Charles M. FULKERSON....b. 10 Jan 1847 in Ohio, m. a Jane _____
Hulbert Luther FULKERSON....b. 24 Jul 1850 in Ohio
Some Disconnected Ohio Fulkersons
Lewis FULKERSON.....Miriam DOWLER on 10 March 1864 in Darke Co., OH
Milton L. FULKESON....m. Catharine MIHSELE on 13 January 1868 in Darke Co., OH
William H. FULKISON....m. Ellen MIKESELL on 29 September 1867 in Darke Co., OH

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{sec:introduction}
With comprehensive application of electromagnetic wave spectrum in radar, communication and navigation and increased computing power and storage, the amount of electromagnetic data comes to a huge number. Less well understood is how to distill the underlying electromagnetic physical laws from electromagnetic data in complex systems. The big electromagnetic data provides new opportunities for the data-driven discovery of new physics laws or making complex system modelling computationally feasible. Traditional derivation of governing equations relies on fundamental laws and theoretical  solution can be obtained with analytic and computational methods. However, the realistic scene in application is complex to tackle, which involves multiple-physics, multiple-scale and nonlinear. For a multiple-physics system with complex interaction mechanisms, there is no exact quantitative analytic solution and computational cost is high to solve a set of partial differential equations (PDEs). Therefore, the key question is how to use sparse data given to discover the principle of a complex system to make predictions of physical quantities outside the range of the measurement data. If the model can be learned from spare data and then perform prediction, it is significant for complex electromagnetic multiple-physics system and beyond.

The data-driver discovery methods have been developing quickly during the past decades. It can be mainly divided into three categories, including symbolic regression, sparse regression and deep learning. One of earlier research on data-driven discovery for free-form natural laws is proposed by Bongard and Lipson (2007) and Schmidt and Lipson (2009) \cite{b1}. The main idea is to calculate numerical differentiation of experimental data first and use symbolic regression based on evolutionary algorithm to compare with analytical derivative solutions. Later, sparse regression method was raised by Brunton et al. (2016) \cite{b3}, Schaeffer (2017)\cite{b4}, and Rudy et al. (2017)\cite{b5}. A candidate function dictionary needs to be established which consists of simple and derivatives terms and then use sparse regression to select the proper terms as part of the equations. Recently, Raissi and Karniadakis (2017) used full connection deep neural network to discover underlying physics of nonlinear PDEs \cite{b6,b7} with less data. The key idea is to set up a universal neural network to approximate the solution of the PDE by minimizing the loss function and the derivatives can then be calculated by automation differentiation based on neural network\cite{b7}. However, the explicit form of the PDEs is assumed to be known. Recently, Long et al. (2018) \cite{b7} utilized the connection between differentiation and convolution to discover nonlinear equations with minor knowledge on the equation form. The wavelet frame filters in convolutional neural network (CNN) with a training kernel is adopted to approximate spatial differentiations. Long et al. (2018) \cite{b8} upgraded their network by imposing appropriate constraints on filters and using a newly designed symbolic neural network to express the analytical form of function clearly.

In sum, the data-driven discovery method of the underlying physics of PDEs are in progress. However, they still have some issues, such as symbolic regression with high computation cost and difficulty for dealing with large scale problem. Sparse regression needs to set numerical differentiations beforehand which costs large storage and may generate unrelated terms. The PDE-net for discovering unknown equations by Long et al. (2018) \cite{b6}\cite{b7} requires no knowledge on the differential operators and associated discrete approximations, which has been applied into 2-dimensional linear variable-coefficient convection-diffusion equation.

However, it is still challenging for the data-driven method in order to solve multiple-physics electromagnetic physics problem. There are two main characteristics for data-driven methods for multiple physics electromagnetic application. First, the electromagnetic scatter field is usually varying fast with time. The characteristic of the EM field needs to be extracted in the time series of data. Secondly,  multiple-physics electromagnetic problems involve a set of coupled different partial differential equations, which may inherit time-varying inhomogeneous coefficients. It requires one unified algorithm to retrieve multiple coefficients of the PDE equation set simultaneously. Therefore, the data-driven discovery method for complex electromagnetic system will be specially designed as a framework.

This paper aims to design a data-driven network architecture to discover a set of nonlinear PDE equations for multiple-physics EM problem. A unified neutral network with CNN is proposed in combination with spare regression and pareto analysis. This proposed network is applied for a electromagnetic wave and plasma interaction system, which can discover the coefficient of the PDE set with relative good accuracy. Also, we investigate the possibility of inferring inhomogeneous coefficients with little prior knowledge. The data-driven methods of the PDEs set will pay the way for deriving equations for complex partially known and unknown systems, including nonlinear, multiple physics, Maxwell's equations equations and beyond.

\section{Methodology}
For simplicity, we assume there are two physical variable $ u(x,y,t), v(x,y,t) $ in a 2-dimensional space $(x,y)$ varying with time $t$. The universal expression of their governing equations can be expressed as following

\begin{spacing}{2.0}
\begin{equation}
\begin{cases}
\dfrac{\partial u}{\partial t} = f(x,y,u,v,\dfrac{\partial u}{\partial x},\dfrac{\partial u}{\partial y},\dfrac{\partial v}{\partial x},\dfrac{\partial v}{\partial y},\dfrac{\partial^{2} u}{\partial x \partial y},\dfrac{\partial^{2} v}{\partial x \partial y}...)\\

\dfrac{\partial v}{\partial t} = g(x,y,u,v,\dfrac{\partial v}{\partial x},\dfrac{\partial v}{\partial y},\dfrac{\partial u}{\partial x},\dfrac{\partial u}{\partial y},\dfrac{\partial^{2} v}{\partial x \partial y},\dfrac{\partial^{2} u}{\partial x \partial y}...)\label{eq}

\end{cases}
\end{equation}
\end{spacing}
where $f$ and $g$ are  nonlinear function of all possible terms including $u$ and $v$, their first and second order derivatives and other parameters, respectively.
\subsection{Time derivative}
We define $ u(t_{i},\cdot) $ or $ v(t_{i},\cdot) $ as all spatial value of function $ u $ or $v$ at $ t = t_{i} $. For time derivative, according to the forward euler method, we obtain the following formula

\begin{equation}
\begin{split}
\tilde{u}(t_{i+1},\cdot)=u(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial u}{\partial t}\\
\tilde{v}(t_{i+1},\cdot)=v(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial v}{\partial t}
\end{split}
\label{eq}
\end{equation}

where $ \tilde{u}(t_{i+1},\cdot) $ or $ \tilde{v}(t_{i+1},\cdot) $ is the approximation value at $ t_{i+1} $.
Then, we obtain expression

\begin{equation}
\begin{split}
\tilde{u}(t_{i+1},\cdot)=u(t_{i},\cdot)+\Delta t\cdot f\\
\tilde{v}(t_{i+1},\cdot)=v(t_{i},\cdot)+\Delta t\cdot g
\end{split}
\label{eq}
\end{equation}

\subsection{Spatial derivative}
For spatial derivative calculation, the connection between convolutions and differentiations was studies by Cai et al. (2012) \cite{b10} and by Dong et al. (2017) \cite{b8}. Here, we demonstrate one simple example of their work \cite{b8}\cite{b10} to show how to express differentiation by convolution. Consider that the 2-dimensional Haar wavelet frame filter banks containing one low-pass filter $ h_{00} $ and three high-pass filters $ h_{10} $, $ h_{01} $ and $ h_{11} $ have forms :

\begin{equation}
\begin{split}
h_{00}=\dfrac{1}{4}\begin{pmatrix}1 & 1\\1 & 1\end{pmatrix},h_{10}=\dfrac{1}{4}\begin{pmatrix}1 & -1\\1 & -1\end{pmatrix},\\
h_{01}=\dfrac{1}{4}\begin{pmatrix}1 & 1\\-1 & -1\end{pmatrix},h_{11}=\dfrac{1}{4}\begin{pmatrix}1 & -1\\-1 & 1\end{pmatrix}.
\end{split}
\label{eq}
\end{equation}

Then, the circular convolution (labeled as $\otimes$ ) between filters and function $u$ for instance is expressed as:
\begin{equation}
\begin{split}
h_{00}\otimes u\approx {u}, \\
h_{10}\otimes u\approx \dfrac{1}{2}\delta_{x}\dfrac{\partial u}{\partial x}, \\h_{01}\otimes u\approx \dfrac{1}{2}\delta_{y}\dfrac{\partial u}{\partial y}, \\h_{11}\otimes u\approx \dfrac{1}{4}\delta_{x}\delta_{y}\dfrac{\partial^{2} u}{\partial x \partial y}.
\end{split}
\label{eq}
\end{equation}

where $ \delta_{x} $ and $ \delta_{y}$ are the spatial size of $x$ and $y$ direction of function $ u $ or $ v $, respectively. \\
Similarly for $v$,
\begin{equation}
\begin{split}
h_{00}\otimes v\approx {v}, \\
h_{10}\otimes v\approx \dfrac{1}{2}\delta_{x}\dfrac{\partial v}{\partial x}, \\h_{01}\otimes v\approx \dfrac{1}{2}\delta_{y}\dfrac{\partial v}{\partial y}, \\h_{11}\otimes v\approx \dfrac{1}{4}\delta_{x}\delta_{y}\dfrac{\partial^{2} v}{\partial x \partial y}.
\end{split}
\label{eq}
\end{equation}

By multiplying constant coefficients, convolution can represent differentiation in neural network effectively. The calculation of the second order differentiation is also mentioned in \cite{b7}, it is ignored since our current work does not involve it  as will be discussed in future.

\subsection{$ \Delta t $ block}

Therefore, we obtain a partial neural network structure from time derivative calculation based on forward Euler's method. For simplicity, the partial neural network architecture will be shown only for variable $u$ and target function $f$, similarly for variable $v$ and target function $g$ .

Figure 1 illustrates a $ \Delta t $ block \cite{b8} as a layer of neural network to advance variables based on equations in time. In principle, the structure of the $\Delta t $ block is an interpretation of Equation (3). The two left terms in Fig.1 represent $\Delta t \cdot f $, that is a candidate terms database in the form of neural network. Thus, the input of this network is clear, that's all possible functions with $ u $ and $ v $ in this example. Here, it is noted that for variable $v$ and its derivatives are also included for advancing $u$ in the candidate database because we assume that there is coupling between $u$ and $v$. The prior knowledge will help reduce the number of possible candidate functions associated with PDEs. The output of $ \delta t $ block is the predicted value at the next time stamp $ \tilde{u}(t_{i+1},\cdot) $.

In general, the whole network consists of several $\Delta t$ blocks by continuously connecting each block one by one. The number of $ \Delta t $ blocks is determined by different convergence criteria.

\begin{figure*}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{01-3.png}
\caption{The architecture of a $ \Delta t $ block \cite{b6} to advance function in time. The two left parts represent the target function $f$ in Equation (3), that includes a candidate term database in the form of neural network as in Equation (5-6). The input of this network is all possibly related functions which are $u$ and $v$ in this example. It is noted that function $v$ and its derivatives are also included in the candidate database because we assume that there exsits coupling between $u $ and $v$ in PDEs. }
\end{figure*}

In Fig. 1, the coefficients $w_{1},w_{2},w_{3}...$ are neural network weights for each candidate term, respectively. The network weights will be shared in each layer and become the output vector of this model. The non-zero weight represents the term of the target equation, where $n$ means the number of candidate terms. We define it $\textbf{W}$ as
\begin{equation}
\textbf{W}= [w_{1}, w_{2}, w_{3}, w_{4}, w_{5},...,w_{n}],
\label{eq}
\end{equation}

\subsection{Sparse regression}
Sparse regression is a common method in data-discovery area. When there are many terms to discover with spare data, it means that the system may face with over-fitting problems. Regularization method need to be adopted in training process and $ L_{1} $ norm regularization is commonly used as $L_{1}$ is easier to calculate than $ L_{0} $ norm regularization.

In our approach, when the training of network is finished, the output $ \textbf{W} $ will be limited by regularization method in the final layer of network.

The output of $ \delta t $ block is the predicted value at the next time step $ \tilde{u}(t_{i+1},\cdot) $. Therefore, the loss function of a single block is defined as
\begin{equation}
l_{i} = \parallel \tilde{u}(t_{i+1},\cdot)-u(t_{i+1},\cdot) \parallel _{2}
\label{eq}
\end{equation}

Consider sparse regression to optimize the output $ \textbf{W} $ at the end of the complete network, we define that loss function of the entire network is sum of each block's loss according to $L_{1}$ norm of $\textbf{W}$. The accumulated loss function can make prediction reliable within certain time step numbers $N$. The expression for final loss $L$ is
\begin{equation}
L = \sum_{i=1}^N \parallel \tilde{u}(t_{i},\cdot)-u(t_{i},\cdot) \parallel _{2}+\lambda\parallel \textbf{W} \parallel_{1}
\label{eq}
\end{equation}
where $\lambda$ is the regularization parameter to determine the weight of regularization term.

\subsection{Pareto analysis}
Pareto analysis is a technique that can be useful in situations where many possible action processes are competing for attention. In essence, the problem-solver evaluates the benefits of each action and selects some of the most effective actions that bring the total benefits reasonably close to the maximum possible benefits. While it is common to refer to pareto as the "80/20" rule, in all cases, 20\% of the reasons for 80\% of the problems are assumed to be a convenient rule of thumb, not and should not be considered an immutable law of nature. This technique helps identify the top priority part of the problem that needs to be addressed.

The application of Pareto analysis adopted in \cite{b6} is the final guarantee of the discovery result of the system. By cutting the term with the minimum parameter out and training again, we compare the later loss with the previous one. If loss reduces, then we repeat the process until loss rises.

Figure 2 illustrates the architecture of the whole neutral network to deal with a given PDE equation set. This process includes
\begin{enumerate}
  \item Find out the number of possible equations to discover, e. g., $f$ or $g$ .
  \item Determine time derivatives, e. g.,$\tilde{u}(t_{i+1},\cdot)$ or $\tilde{v}(t_{i+1},\cdot)$.
  \item Decide the number of networks to activate.
  \item Utilize spare regression to train network weights $\textbf{W}$.
  \item Cut out terms based on Pareto analysis.
  \item Iterate and train again if not converged.
\end{enumerate}

\begin{figure*}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{02-3.png}
\caption{The architecture of discovering the PDE equation set. The procedure steps include 1) Check the number of PDEs equations. 2) Determine the time derivative of active terms in the PDE. 3) Activate multiple equation network corresponding to each equations. 4) Train network with sparse regression. 5) Pareto analysis helps to cut out unrelated terms with minimum parameters.}
\end{figure*}

\section{Simulation}
\subsection{Forward model}
Here, we consider the interaction between electromagnetic waves and magnetized plasmas with collisions. For such multiple physic complex system, the Maxwell's partial differential equations and constitutive relations are given as

\begin{equation}
\begin{split}
\nabla\times\textbf{H}=\varepsilon_{0}\dfrac{\partial \textbf{E}}{\partial t}+\textbf{J}\\
\nabla\times\textbf{E}=-\mu_{0}\dfrac{\partial \textbf{H}}{\partial t}\\
\dfrac{\partial \textbf{J}}{\partial t}+\nu_{c}\textbf{J}=\varepsilon_{0}\omega_{p}^{2}\textbf{E}+\omega_{ce}\times\textbf{J}
\end{split}
\end{equation}

where ${\mathbf{H}}$ , ${\mathbf{E}}$ and ${\mathbf{J}}$ are magnetic intensity vector, electric field intensity vector, and polarized current density vector respectively. ${\varepsilon _0}$, ${\mu _0}$ are the vacuum permittivity and magnetic permeability respectively. Here, ${\omega _p}$, ${\omega _{\rm{ce}}}$ and ${v_{c}}$ are plasma frequency, electron cyclotron frequency and electron collision frequency, respectively.
The background magnetic field is $\mathbf{B_{0}} = B_{y}\hat{y} + B_{z}\hat{z}$. The wave propagates with magnetic inclination angle ${\theta}$ with respect to the ${\mathbf{z}}$ axis. Equation (3) is only for cold plasmas and the ion motion is neglected.

To verify our method, we calculate the scattering field $\mathbf{E_{\rm s}}$ for a one-dimensional magnetized plasma slab shown in Fig.3. The computational space takes up 800 grid along the ${\mathbf{z}}$ directions, among which plasma takes up 200-600 grids with uniform distribution and a thickness of $d =3 \, \rm{cm}$. The JEC-FDTD method is applied to obtain the solution of Equation(9). The algorithm can be found in details \cite{b11}. For uniform plasma distribution, parameters in the simulation are set as following in Table I.

\begin{table}[h]
\caption{System Parameter}
\label{table}
\center
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|p{150pt}|p{50pt}|}
\hline
\rule{0pt}{10pt}
Symbol &
Value \\
\hline
\rule{0pt}{10pt}
Electron collision frequency $\nu_{c}$&
$  20 $ GHz  \\
\hline
\rule{0pt}{10pt}
Electron cyclotron frequency $\omega_{ce}$&
$  30 $ GHz  \\
\hline
\rule{0pt}{10pt}
Uniform plasma frequency $\omega_{p}$&
$  40 $ GHz  \\
\hline
\rule{0pt}{10pt}
Magnetic inclination angle $\theta$ &
$ 90\,\degree $ \\
\hline
\end{tabular}
\label{tab1}
\end{table}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{03-3.png}
\caption{Multiple-physics system: EM wave interaction with plasmas  with arbitrary magnetic inclination $\theta$. The spatial domain is in the ${\mathbf{z}}$ direction.}
\end{figure}

For $\theta= \pi/2$, Equations (10) are expressed as a scalar differential equation as:
\begin{equation}
\dfrac{\partial E_{x}}{\partial z}=-\mu_{0}\dfrac{\partial H_{y}}{\partial t}
\end{equation}
\begin{equation}
-\dfrac{\partial H_{y}}{\partial z}=\varepsilon_{0}\dfrac{\partial E_{x}}{\partial t}+J_{x}
\end{equation}
\begin{equation}
\dfrac{\partial J_{x}}{\partial t}+\nu_{c}J_{x}=\varepsilon_{0}\omega_{p}^{2}E_{x}+\omega_{ce}J_{z}
\end{equation}
\begin{equation}
\dfrac{\partial J_{z}}{\partial t}+\nu_{c}J_{z}=\varepsilon_{0}\omega_{p}^{2}E_{z}-\omega_{ce}J_{x}
\end{equation}
\begin{equation}
J_{z}=-\varepsilon_{0}\dfrac{\partial E_{z}}{\partial t}
\end{equation}

Here, there are 5 Equations to discover with five unknown parameters $\mu_{0}, \varepsilon_{0}, \varepsilon_{0}\omega_{p}^{2}, \omega_{ce}, \nu_{c}$. This set of PDE equations is our first goal for equation discovery. This set of PDE equations is for EM wave propagation in anisotropic plasmas, however, it can effective reveal the application for our proposed network.

\subsection{Prepossessing}
The whole process of inverse model is shown in Fig. 4. First of all, the physical input data are $E_{x}, J_{x}, E_{z}, J_{z}, H_{y}$ calculated from JEC-FDTD algorithm. Then, sampling is conducted for the input data $ E_{x}, J_{x}, E_{z}, J_{z}, H_{y}$ for training, respectively.

Next, normalization is executed directly after data sampling. As we use the same input for different equations, we need to do normalization for each equations because the order of magnitudes of input data will cause big error for data training and lead to wrong results. Specifically, we multiply each input data with a designed parameter respectively to keep them into the same order of magnitude. Afterwards, each input data should be compared with time derivative $\frac{\partial E_{x}}{\partial t},\frac{\partial J_{x}}{\partial t},\frac{\partial E_{z}}{\partial t},\frac{\partial J_{z}}{\partial t}, \frac{\partial H_{y}}{\partial t}$, which are included by the target equation to keep in the same magnitude.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.3\textwidth]{04-4.png}
\caption{The overall process of the inverse model for a multiple-physics EM problem. Step 1: Data collected from a solution of PDE Equations (11-15). Step 2: Data sparse sampling and normalization. Step 3: Check the  number of equations and determine the time derivative of the input data. Step 4: Multi-Equation network activation and training shown in Fig.1. Step 5: Sparse regression is set at the end of training and the detailed form of equations are shown. Step 6: After Pareto analysis, distortion terms will be deleted.}
\end{figure}

For collecting training data  we select three physical quantity of the scattered field $ E_{x}, E_{z}, H_{y}$ and two currents $J_{x}, J_{z}$ at spatial grids $z(\Delta z)=190\sim210$ including plasma-vacuum boundary (boundary grid $200$). For temporal series of physical quantities, we choose few samples at the time period $t(\Delta t)=500\sim510$, which includes the maximum value of the scattered field from plasmas.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.50\textwidth]{fig3.png}
\caption{Data collection location - for upper part, this is a x-direction electric field intensity $ E_{x} $ picture. The two white lines means the incident location and exit location. The small white square represents the collection area. For lower part, it is the all points collected, where the grey line represents the incident border of plasma. We captured points both inside and outside plasma to ensure the reliability.}
\end{figure}

Figure 5 shows examples for the total electric field $E_{x}$ and $ E_{z}$ in perpendicular to the magnetic field. The sampling scheme is the same for all five quantities. In Figure 5, the two white lines represent the boundary of plasma. The small white square represents the data collection area from both inside and outside the plasma. For time sampling, we choose 10 time steps after time 500, because the gradient of curve at this time is obvious so that the function is easy to be recognized by neural network.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.50\textwidth]{fig4.png}
\caption{Sampling area - in the upper panel, the black square shows the data sampling area for $E_{x}$ and $E_{z}$ in time domain. In the bottom panel, the black horizontal line indicates the border of plasma and our collection area begins right on the incident border.}
\end{figure}

The sampling area for training network is shown in Figure 6. In the upper panel, the black square shows the data sampling area for $E_{x}$ and $E_{z}$ in time domain. In the bottom panel, the black horizontal line indicates the border of plasma and our collection area begins right on the incident border. It shows that only a small number of time and spatial information are continuously sampled for training.

\subsection{Network Construction}
From forward Euler's method, we have
\begin{equation}
\tilde{E_{x}}(t_{i+1},\cdot)=E_{x}(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial E_{x}}{\partial t}
\end{equation}
\begin{equation}
\tilde{H_{y}}(t_{i+1},\cdot)=H_{y}(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial H_{y}}{\partial t}
\end{equation}
\begin{equation}
\tilde{J_{x}}(t_{i+1},\cdot)=J_{x}(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial J_{x}}{\partial t}
\end{equation}
\begin{equation}
\tilde{J_{z}}(t_{i+1},\cdot)=J_{z}(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial J_{z}}{\partial t}
\end{equation}
\begin{equation}
\tilde{E_{z}}(t_{i+1},\cdot)=E_{z}(t_{i},\cdot)+\Delta t\cdot\dfrac{\partial E_{z}}{\partial t}
\end{equation}

For one-dimension problem, the spatial direction is only $ z $, so we use only one filter to get one kind of spatial differentiation. Take $ E_{x} $ for example:
\begin{equation}
h_{10}\otimes E_{x}\approx \dfrac{1}{2}\delta_{z}\dfrac{\partial E_{x}}{\partial z}
\end{equation}

Based on this, we get the adjusted neural network structure. To fulfil the function of Equation (21), we use a funtion called conv2d in TensorFlow to generate convolution calculation. In our case , since it is one dimensional 1D problem, we obtain the shape of convolution kernel of $ 2\times1 $, so that this function still works.

By screening different weights of network, the correct equation term and its corresponding coefficient can be obtained. This is one single time block for Equations (11-15).

As one single block $\delta_{t}$ given above, we have 10 $\delta_{t}$ time block to build up the whole network. By using the parameter sharing characteristic of neural network, the training coefficient results become accurate if the number of time blocks is increased. Consider that all redundant terms may lead to the problem of complicated neural network structure, we reduce the number of candidates based on certain prior physical knowledge, thus making the network training more efficient.

\begin{table*}[htbp]
\caption{Inversion Result for PDEs with 10 spatial points for each equation}
\label{table}
\centering
\setlength{\tabcolsep}{5pt}
\begin{tabular}{|p{130pt}|p{170pt}|p{170pt}|}
\hline
\rule{0pt}{15pt}
Equation&
Error without noise&
Error with noise SNR = 40$\rm{dB}$ \\
\hline
\rule{0pt}{18pt}
$ \dfrac{\partial E_{x}}{\partial z}=-\mu_{0}\dfrac{\partial H_{y}}{\partial t} $ &
$ \mu_{0}: 1.68\% $

&
$ \mu_{0}: 10.39\% $ \\
\hline
\rule{0pt}{18pt}
$ -\dfrac{\partial H_{y}}{\partial z}=\varepsilon_{0}\dfrac{\partial E_{x}}{\partial t}+J_{x} $ &
$ \varepsilon_{0}:10.14\% $ (the average of two terms)

&
$ \varepsilon_{0}:12.92\% $ (the average of two terms) \\
\hline
\rule{0pt}{18pt}
$ \dfrac{\partial J_{x}}{\partial t}+\nu_{c}J_{x}=\varepsilon_{0}\omega_{p}^{2}E_{x}+\omega_{ce}J_{z} $ &
$ \nu_{c}: 9.12\%,\: \varepsilon_{0}\omega_{p}^{2}: 0.53\%,\: \omega_{ce}: 1.06\% $
&
$ \nu_{c}: 23.38\%,\: \varepsilon_{0}\omega_{p}^{2}: 6.84\%,\: \omega_{ce}: 3.07\% $\\
\hline
\rule{0pt}{18pt}
$\dfrac{\partial J_{z}}{\partial t}+\nu_{c}J_{z}=\varepsilon_{0}\omega_{p}^{2}E_{z}-\omega_{ce}J_{x} $ &
$ \nu_{c}: 0.39\%,\: \varepsilon_{0}\omega_{p}^{2}: 1.02\%,\: \omega_{ce}: 1.46\% $
&
$ \nu_{c}: 9.19\%,\: \varepsilon_{0}\omega_{p}^{2}: 5.73\%,\: \omega_{ce}: 35.14\% $
\\
\hline
\rule{0pt}{18pt}
$ J_{z}=-\varepsilon_{0}\dfrac{\partial E_{z}}{\partial t} $ &
$ \varepsilon_{0}:4.32\% $  &
$ \varepsilon_{0}:5.33\% $
\\
\hline
\end{tabular}
\label{tab1}
\end{table*}

\subsection{Result and Analysis}
For the system of PDEs set, we build a neural network based on sparse temporal-spatial sample data for training and predict the coefficients of PDEs without interfering each other. In this process, we need to fully normalize the data, taking into account that the magnitude of various field quantities of electromagnetic field data with huge differences. By evaluating the order of magnitude of each candidate, we set a reasonable normalization coefficient for each candidate with certain prior knowledge to ensure that the disparity of magnitude order will not cause great impact on training results of the network.

\subsubsection{Homogeneous simulation}
Wave propagation in homogeneous medium is the most common problem for EM applications. Therefore, we select a homogeneous case as our first attempt to validate methodology proposed above.

Table II summarizes inversion results of proposed methodology applied to a multiple-physics EM interaction system of Equations (11-15) with all coefficients homogeneous. All data shown in Table II are sampled from $ z=200 $ to $ z=209 $, and $ t=500 $ to $ t=509 $. In total, 10 spatial points and 10 time points are selected for comparison. The five PDEs are obtained simultaneously with five coefficients $\mu_{0}, \varepsilon_{0},\nu_{c}, \varepsilon_{0}\omega_{p}^{2}, \omega_{ce}$. It is noted that $\nu_{c},\omega_{p}, \omega_{ce}$ are three important  parameters of collisional magnetized plasmas.  For simplicity, we combine $ \epsilon_{0}\omega^{2}_{p} $ together as one parameter. Most of the relative errors are relatively low approximately less than $5\%$. We add noise into the input data with signal to noise ratio $\rm{SNR}=40 \,\rm{dB}$. The equations with some noise $\rm{SNR}=40 \,\rm{dB}$ can also be identified with increased errors. In general, the inversion error increases but can still discover physical system even if significantly sub-sampled spatially and temporally.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{fig7.png}
\caption{Inversion accuracy dependent on spatial sampling numbers. It is noted that each point indicates the accuracy of normalized parameters. The spatial sampling points are 2, 3, 5, 10, 15, 20, respectively and the beginning location is $z=200 $. It is noted that inversion accuracy reaches to $95\%$ only for three samples but reduces substantially for two samples.}
\end{figure}

The sampling scheme of physical quantities in spatial and time domain is the most important parameter for calculating derivatives to ensure inversion accuracy. On one hand, the dependence of inversion accuracy on spatial sampling numbers is depicted for coefficients $\nu_{c}, \varepsilon_{0}\omega_{p}^{2}, \omega_{ce}, \varepsilon_{0}, \mu_{0}$ in PDEs in Figure 7. The spatial sampling points are 2, 3, 5, 10, 15, 20, respectively and the beginning location is $z=200 $ at  plasma-vacuum boundary. It is noted that inversion accuracy reaches to $95\%$ for three spatial sampling position but reduces substantially for just two spatial samples. Thus, for considering the spatial sampling limitation, we need to make trade-off between accuracy and spatial sampling scale.

On the other hand, temporal sampling scenario of the field sequence data plays an important role in inversion accuracy as well. Figure 8 depicts the dependence of accuracy $\varepsilon_{0}$ on the gradient of $E_{\rm z}$ based on Equation (15) as an example. We select three typical time sampling blocks corresponding to low (red), middle (blue), and high gradient (green) in the upper panel of Figure 8. In each block, there are 10 temporal samples. It is noted that the accuracy reaches $99\%$ for high gradient (green) and reduces to $90\%$ for low gradient (red). It is noted that the inversion accuracy correlates with the time derivative of the sampled data.  With data varying rapidly with time, the neutral network can catch  data characteristics easily. For complete PDE equation set, it will require a common time block for all quantities with high gradients within reasonably large amplitude.

\begin{figure}[htbp]

\includegraphics[width=0.5\textwidth]{fig8.png}
\caption{Inversion accuracy dependent on temporal sampling scenario. Three time sampling blocks are marked for low (red), middle (blue), and high gradient (green) with 10 samples in each block. Here is an example of Equation (15) with the time derivative of $ E_{z} $ at spatial location $ z=200 $. For low gradient (red block in upper panel), the accuracy is low (red in lower panel). It is noted that the inversion accuracy correlates with the time derivative of sampled data block.  }
\end{figure}

\subsubsection{Inhomogeneous simulation}
For all simulation above, we assume homogeneous coefficient for discovering the PDEs set. To verify algorithm further, we investigate the inhomogeneous coefficient further which often occurs for a wide range of applications.

Network adjustment. The structure of neural network needs to be adjusted to solve inhomogeneous problems. First, as the coefficients of inhomogeneous terms are varying with spatial position, we change parameters of inhomogeneous terms from a single number to a length-adaptive tensor. We call this length-adaptive tensor \textit{inhomogeneous parameter tensor}. This tensor will be trainable in neural network. The length of inhomogeneous parameter tensor is equal to the spatial sampling point number. Secondly, we assume that prior knowledge is known which candidate terms have inhomogeneous parameters so that not all candidate terms should have inhomogeneous parameter tensors.
Finally, the predicted parameter is no longer a single number but a tensor which indicates the varying parameters within spatial domain. After training, the underlying equations with inhomogeneous coefficients can be distilled from the inhomogeneous parameter tensor by the gradient descent method.

The input data of the scattered field for inhomogeneous density  is obtained based on the JEC-FDTD algorithm. Here, we assume that $\omega_{p}^{2}$ obeys a sinusoidal envelope, positively proportional to plasma density for inversion. A normal expression of varying parameter can be written as
\begin{equation}
\omega_{p}^{2}\sim A\sin(kz+\varphi)+C
\end{equation}
Here, $A, k, \varphi, C$ is amplitude, wave number, phase and constant, respectively. Plasma parameter $\omega_{p}^{2}$ varies as Equation (22) from $z=200$ to $z=600$ with two wavelength inside plasmas correspondingly.

In our simulation, the exact form and normal expression of inhomogeneous parameter are unknown. In general, we assume that unknown inhomogeneous coefficients can be expressed as a kind of trigonometric series in the form of
\begin{equation}
\dfrac{1}{2}A_{0}+\sum_{n=1}^{N}(A_{n}\sin (nx)+B_{n}\cos (nx))
\end{equation}
where $n$ is the order of the series, $A_{n}$ and $B_{n}$ are coefficients, respectively.

We assume that $ N = 3 $ is sufficient to represent the expression of inhomogeneous parameter tensor. Thus, the estimated function of inhomogeneous parameters  $\tilde{h}(z)$ can be expressed as
\begin{equation}
\begin{split}
\tilde{h}(z)=A_{1}\sin (kz) + B_{2}\cos (kz) + A_{2}\sin (2kz) +\\
B_{2}\cos (2kz) + A_{3}\sin (3kz) + B_{3}\cos (3kz) + C
\end{split}
\end{equation}
where $A_{1}, A_{2}, A_{3}$ and $B_{1}, B_{2}, B_{3}$ are unknown coefficients to discover, respectively.

All unknown coefficients are determined in Equation (24) by  sparse regression. We use adaptive moment optimizer (Adam optimizer) to solve the following equation
\begin{equation}
\begin{split}
\arg\min_{\mathbf{S}}\sum\Vert h(z)-\tilde{h}(z)\Vert_{2}+\lambda\Vert{\mathbf{S}}\Vert_{1} \\
{\mathbf{S}}=[A_{1}, B_{1}, A_{2}, B_{2}, A_{3}, B_{3}, C]
\end{split}
\end{equation}

For spatial sampling, we choose 20 spatial position collected from $z=240-260$ for training and then predict inhomogeneous coefficient in the whole spatial domain. The identified coefficients of ${\mathbf{S}}$ are $A_{1}=0.5036, B_{1}=-0.0011$, $A_{2}=-0.0020, B_{2}=0.0002$ and $A_{3}=0.0005, B_{3}=0.0018$, respectively. The identified constant number is $C=0.9951$ with true value 1. The dominant term $A_{1}\sin(kz)$ is successfully selected among all candidates and other terms are small enough to be ignored. The true and identified models of inhomogeneous coefficient are given by
\begin{equation}
\omega_{p}^{2}\sim 0.5\sin(3.142z)+1 \,\rm{(True)}
\end{equation}
\begin{equation}
\omega_{p}^{2}\sim 0.5036\sin(3.123z)+0.9951\, \rm{( Identified)}
\end{equation}

Figure 9 shows the inversion result of inhomogeneous coefficient in comparison with theoretical value by Equation (26). The spatial sampling position is shown by the black box from $z=240-260$ for training. Although sampling only at few spatial grids, inhomogeneous coefficients can be obtained in the sampling regime and then make prediction outside measurement regime according to discovered equations . Basically, the inversion result agrees well with theoretical value after iteration with only 20 spatial and 10 temporal samples. For a Gaussian white noise
of $\rm{SNR}=40\rm{dB} $, the identified model with noise is
\begin{equation}
\omega_{p}^{2}\sim 0.5897\sin(3.360z)+1.082\, \rm{( Identified)}
\end{equation}

However, there are some errors of prediction with noise. One possible source of error comes from the neural network training of inhomogeneous parameter tensor. Another source of error may arise from processes during distilling the underlying equation from inhomogeneous parameter tensor. Even though, this method still discover and predict the inhomogeneous parameter.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{fig9.png}
\caption{Inhomogeneous parameter prediction without noise. The blue solid line is theoretical value, and the dotted line is predicted value. The black box indicates spatial sampling position from $z=240-260$ inside plasmas for training.}
\end{figure}

\section{Conclusion and Discussion}
In this paper, we have presented a a data-driven network architecture to discover the hidden nonlinear PDE equation set, where first-principles derivation may be intractable. The architecture extends the idea from the PDE-net with compressive sensing, which approximates differential operations by convolutions with properly constrained filters and approximate the nonlinear response by deep neural networks. We add a coefficient regression module to the system and Pareto analysis to improve the identification accuracy. The architecture can contribute construct a model which characterizes the observed dynamics and generalizes to unsampled regime of parameter space.

We apply this network architecture for Maxwell's equations for a multiple-physics EM system. The discovery of PDEs are complex in terms of multiple-physics, non-linearity, inhomogeneity, and anisotropy. We can discover Maxwell's equations coupled with plasma current conservation equation with low error on the coefficient of PDEs. The homogeneous coefficients of PDEs have been constructed based on sparse temporal-spatial data. Also, our attempt demonstrates the possibility of inferring inhomogeneous coefficients of physical parameters with prior knowledge. By assuming that the expression of coefficient of inhomogeneous parameters can be equal to some order of trigonometric series, we use sparse regression to discover the equation underlying in the inhomogeneous coefficients with little prior knowledge. The identified equation requires a batch of adjacent temporal-spatial samples to evaluate derivatives of physical variables. When the discretized solution contains measurement noise, the numerical derivatives becomes challenging to evaluate that will need further work.

Finally, it is noted that current network has been tested for the 1-D PDEs  for the extraordinary wave $\theta=90^{\rm{o}}$ with only 5 equations coupled. Considering arbitrary angle $\theta$ in 3-D spatial domain, we need to add the representation module of curl operator in the neural network. Secondly, this method has been simplified to identify models with some prior knowledge for inhomogeneous parameter. If no prior knowledge is available, we will need a new candidate function database for identifying unknown terms similary as homogenous network. Thirdly, kinetic effects of plasmas are ingnored in the forward model, and we make assumputions for cold plasmas as well. Finnaly, it is valuable to try the proposed framework on real data of nonlinear EM wave and space plasma interaction experiments\cite{b12,b13,b14} and other potential applications.

\section*{Acknowledgement}
This work was supported by the National Key Research and Development Program of China under Grant 2017YFB0502703 and also supported by the Shanghai Science Foundation under Grant 19ZR1403900 and Grant 19511100600.

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