Recursive Regression with Neural Networks: Approximating the HJI PDE Solution
Most machine learning applications using neural networks seek to approximate some function g(x) by minimizing some cost criterion. In the simplest case, if one has access to pairs of the form (x, y) where y = g(x), the problem can be framed as a regression problem. Beyond this family of problems, we find many cases where the unavailability of data pairs makes this approach unfeasible. However, similar to what we find in the reinforcement learning literature, if we have some known properties of the function we are seeking to approximate, there is still hope to frame the problem as a regression problem. In this context, we present an algorithm that approximates the solution to a partial differential equation known as the Hamilton-Jacobi-Isaacs PDE and compare it to current state of the art tools. This PDE, which is found in the fields of control theory and robotics, is of particular importance in safety critical systems where guarantees of performance are a must.
This paper presents an algorithm for approximating the solution of certain time-evolution PDEs. The paper presents an interesting learning-based approach to solve such PDEs. The idea is to alternate between: 1. sampling points in space-time 2. generating solution to PDE at *those* sampled points 3. regressing a space-time function to satisfy the latter solutions at the sampled points (and hopefully generalize beyond those points). I actually find the proposed algorithm interesting, and potentially useful in practice. The classic grid-based simulation of PDEs is often too expensive to be practical, due to the curse of dimensionality. Hence, learning the solution of PDEs makes a lot of sense for practical settings. On the other hand, as the authors point out, simply running gradient descent on the regression loss function does not work, because of the non-differentiablity of the *min* that shows up in the studied PDEs. Therefore, I think the proposed idea is actually very interesting approach to learning the PDE solution in presence of non-differentability, which is indeed a *challenging* setup for numerically solving PDEs. The paper motivates the problem (time-evolution PDE with *min* operator applied to the spatial derivatives) by applications in control thery, but I think there is more direct interest in such problems for the machine learning community, and even deep learning community. For example http://link.springer.com/chapter/10.1007/978-3-319-14612-6_4 studies approximate solution to PDEs with very similar properties (evolution+*min*) to develop new optimization algorithms. The latter is indeed used to training deep networks: https://arxiv.org/abs/1601.04114 I think this work would catch even more attention if the authors could show some experiments with higher-dimensional problems (where grid-based methods are absolutely inapplicable).
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Recursive Regression with Neural Networks: Approximating the HJI PDE Solution
Most machine learning applications using neural networks seek to approximate some function g(x) by minimizing some cost criterion. In the simplest case, if one has access to pairs of the form (x, y) where y = g(x), the problem can be framed as a regression problem. Beyond this family of problems, we find many cases where the unavailability of data pairs makes this approach unfeasible. However, similar to what we find in the reinforcement learning literature, if we have some known properties of the function we are seeking to approximate, there is still hope to frame the problem as a regression problem. In this context, we present an algorithm that approximates the solution to a partial differential equation known as the Hamilton-Jacobi-Isaacs PDE and compare it to current state of the art tools. This PDE, which is found in the fields of control theory and robotics, is of particular importance in safety critical systems where guarantees of performance are a must.
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>



This paper presents an algorithm for approximating the solution of certain time-evolution PDEs. The paper presents an interesting learning-based approach to solve such PDEs. The idea is to alternate between: 1. sampling points in space-time 2. generating solution to PDE at *those* sampled points 3. regressing a space-time function to satisfy the latter solutions at the sampled points (and hopefully generalize beyond those points). I actually find the proposed algorithm interesting, and potentially useful in practice. The classic grid-based simulation of PDEs is often too expensive to be practical, due to the curse of dimensionality. Hence, learning the solution of PDEs makes a lot of sense for practical settings. On the other hand, as the authors point out, simply running gradient descent on the regression loss function does not work, because of the non-differentiablity of the *min* that shows up in the studied PDEs. Therefore, I think the proposed idea is actually very interesting approach to learning the PDE solution in presence of non-differentability, which is indeed a *challenging* setup for numerically solving PDEs. The paper motivates the problem (time-evolution PDE with *min* operator applied to the spatial derivatives) by applications in control thery, but I think there is more direct interest in such problems for the machine learning community, and even deep learning community. For example http://link.springer.com/chapter/10.1007/978-3-319-14612-6_4 studies approximate solution to PDEs with very similar properties (evolution+*min*) to develop new optimization algorithms. The latter is indeed used to training deep networks: https://arxiv.org/abs/1601.04114 I think this work would catch even more attention if the authors could show some experiments with higher-dimensional problems (where grid-based methods are absolutely inapplicable).
assistant
This paper presents an algorithm for approximating the solution of certain time-evolution PDEs. The paper presents an interesting learning-based approach to solve such PDEs. The idea is to alternate between: 1. sampling points in space-time 2. generating solution to PDE at *those* sampled points 3. regressing a space-time function to satisfy the latter solutions at the sampled points (and hopefully generalize beyond those points). I actually find the proposed algorithm interesting, and potentially useful in practice. The classic grid-based simulation of PDEs is often too expensive to be practical, due to the curse of dimensionality. Hence, learning the solution of PDEs makes a lot of sense for practical settings. On the other hand, as the authors point out, simply running gradient descent on the regression loss function does not work, because of the non-differentiablity of the *min* that shows up in the studied PDEs. Therefore, I think the proposed idea is actually very interesting approach to learning the PDE solution in presence of non-differentability, which is indeed a *challenging* setup for numerically solving PDEs. The paper motivates the problem (time-evolution PDE with *min* operator applied to the spatial derivatives) by applications in control thery, but I think there is more direct interest in such problems for the machine learning community, and even deep learning community. For example http://link.springer.com/chapter/10.1007/978-3-319-14612-6_4 studies approximate solution to PDEs with very similar properties (evolution+*min*) to develop new optimization algorithms. The latter is indeed used to training deep networks: https://arxiv.org/abs/1601.04114 I think this work would catch even more attention if the authors could show some experiments with higher-dimensional problems (where grid-based methods are absolutely inapplicable).