Meta-Learning Runge-Kutta
Initial value problems, i.e. differential equations with specific, initial conditions, represent a classic problem within the field of ordinary differential equations(ODEs). While the simplest types of ODEs may have closed-form solutions, most interesting cases typically rely on iterative schemes for numerical integration such as the family of Runge-Kutta methods. They are, however, sensitive to the strategy the step size is adapted during integration, which has to be chosen by the experimenter. In this paper, we show how the design of a step size controller can be cast as a learning problem, allowing deep networks to learn to exploit structure in the initial value problem at hand in an automatic way. The key ingredients for the resulting Meta-Learning Runge-Kutta (MLRK) are the development of a good performance measure and the identification of suitable input features. Traditional approaches suggest the local error estimates as input to the controller. However, by studying the characteristics of the local error function we show that including the partial derivatives of the initial value problem is favorable. Our experiments demonstrate considerable benefits over traditional approaches. In particular, MLRK is able to mitigate sudden spikes in the local error function by a faster adaptation of the step size. More importantly, the additional information in the form of partial derivatives and function values leads to a substantial improvement in performance. The source code can be found at https://www.dropbox.com/sh/rkctdfhkosywnnx/AABKadysCR8-aHW_0kb6vCtSa?dl=0
The paper proposes to learn the step size for a Runge-Kutta numerical integrator for solving ordinary differential equations initial value problems. The authors frame the stepsize control problem as a learning problem, based on different performance measures, on ODE dependent inputs and on a LSTM for predicting the next step coefficient. Experiments are performed on 3 ODEs by training and testing in different contexts. The problem of designing adaptive controllers for ODE numerical schemes is interesting and is probably a new application for ML. The paper makes an effort to introduce the necessary background and for reviewing some classical adaptive controller techniques. The description of the method is relatively clear, but could however be largely improved in order to make it more accessible to the audience. Many of the arguments and proposed ideas come without justification, some definitions should be made more precise. The construction of the training and test sets should be better explained. The experiments show that the proposed approach leads to fewer evaluations but larger mean errors. The graphics also show that the local error is smaller for the proposed method than for the baselines which is in contradiction with the global error behavior. This should be clarified – the relations between the two error types should be made clear. The baseline is not defined in the text so that it is difficult to judge the performance. Why not comparing to several adaptive baselines? In conclusion, this is an interesting topic, the paper proposes new ideas. A more careful writing and especially a better comparison with sota baselines would greatly improve the paper. ------ post rebuttal ----- Thanks for the answers. I still think that the ideas are interesting but that the experiments do not demonstrate enough of the proposed method. I will keep my score.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Meta-Learning Runge-Kutta
Initial value problems, i.e. differential equations with specific, initial conditions, represent a classic problem within the field of ordinary differential equations(ODEs). While the simplest types of ODEs may have closed-form solutions, most interesting cases typically rely on iterative schemes for numerical integration such as the family of Runge-Kutta methods. They are, however, sensitive to the strategy the step size is adapted during integration, which has to be chosen by the experimenter. In this paper, we show how the design of a step size controller can be cast as a learning problem, allowing deep networks to learn to exploit structure in the initial value problem at hand in an automatic way. The key ingredients for the resulting Meta-Learning Runge-Kutta (MLRK) are the development of a good performance measure and the identification of suitable input features. Traditional approaches suggest the local error estimates as input to the controller. However, by studying the characteristics of the local error function we show that including the partial derivatives of the initial value problem is favorable. Our experiments demonstrate considerable benefits over traditional approaches. In particular, MLRK is able to mitigate sudden spikes in the local error function by a faster adaptation of the step size. More importantly, the additional information in the form of partial derivatives and function values leads to a substantial improvement in performance. The source code can be found at https://www.dropbox.com/sh/rkctdfhkosywnnx/AABKadysCR8-aHW_0kb6vCtSa?dl=0
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



The paper proposes to learn the step size for a Runge-Kutta numerical integrator for solving ordinary differential equations initial value problems. The authors frame the stepsize control problem as a learning problem, based on different performance measures, on ODE dependent inputs and on a LSTM for predicting the next step coefficient. Experiments are performed on 3 ODEs by training and testing in different contexts. The problem of designing adaptive controllers for ODE numerical schemes is interesting and is probably a new application for ML. The paper makes an effort to introduce the necessary background and for reviewing some classical adaptive controller techniques. The description of the method is relatively clear, but could however be largely improved in order to make it more accessible to the audience. Many of the arguments and proposed ideas come without justification, some definitions should be made more precise. The construction of the training and test sets should be better explained. The experiments show that the proposed approach leads to fewer evaluations but larger mean errors. The graphics also show that the local error is smaller for the proposed method than for the baselines which is in contradiction with the global error behavior. This should be clarified – the relations between the two error types should be made clear. The baseline is not defined in the text so that it is difficult to judge the performance. Why not comparing to several adaptive baselines? In conclusion, this is an interesting topic, the paper proposes new ideas. A more careful writing and especially a better comparison with sota baselines would greatly improve the paper. ------ post rebuttal ----- Thanks for the answers. I still think that the ideas are interesting but that the experiments do not demonstrate enough of the proposed method. I will keep my score.
assistant
The paper proposes to learn the step size for a Runge-Kutta numerical integrator for solving ordinary differential equations initial value problems. The authors frame the stepsize control problem as a learning problem, based on different performance measures, on ODE dependent inputs and on a LSTM for predicting the next step coefficient. Experiments are performed on 3 ODEs by training and testing in different contexts. The problem of designing adaptive controllers for ODE numerical schemes is interesting and is probably a new application for ML. The paper makes an effort to introduce the necessary background and for reviewing some classical adaptive controller techniques. The description of the method is relatively clear, but could however be largely improved in order to make it more accessible to the audience. Many of the arguments and proposed ideas come without justification, some definitions should be made more precise. The construction of the training and test sets should be better explained. The experiments show that the proposed approach leads to fewer evaluations but larger mean errors. The graphics also show that the local error is smaller for the proposed method than for the baselines which is in contradiction with the global error behavior. This should be clarified – the relations between the two error types should be made clear. The baseline is not defined in the text so that it is difficult to judge the performance. Why not comparing to several adaptive baselines? In conclusion, this is an interesting topic, the paper proposes new ideas. A more careful writing and especially a better comparison with sota baselines would greatly improve the paper. ------ post rebuttal ----- Thanks for the answers. I still think that the ideas are interesting but that the experiments do not demonstrate enough of the proposed method. I will keep my score.