Title: Compute-Optimal Solutions for Acoustic Wave Equation Using Hard-Constraint PINNs
Venue: NeurIPS 2024
Accepted: no
Keywords: compute-optimal

acoustic wave equation

hard-constraint

PINN
Abstract: This paper explores the optimal imposition of hard constraints, strategic sampling of PDEs, and computational domain scaling for solving the acoustic wave equation within a specified computational budget. First, we derive a formula to systematically enforce hard boundary and initial conditions in Physics-Informed Neural Networks (PINNs), employing continuous functions within the PINN ansatz to ensure that these conditions are satisfied. We demonstrate that optimally selecting these functions significantly enhances the convergence of the solution. Secondly, we introduce an importance sampling strategy that optimizes the efficiency of hard-constraint PINNs under a fixed number of sampling points. Leveraging these strategies, we develop an algorithm to determine the optimal computational domain size, given a computational budget. Our approach offers a practical framework for domain decomposition in large-scale implementation of acoustic wave equation systems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper attempts to train PINNs which solves acoustic wave equations. They do so by using hard-constrained PINNs which can enforce IC and BCs, and propose a collocation point sampling method (DAFS) based on the amplitude of the solution at different regions.\n\nSTRENGTHS:\nThe paper considers an interesting problem in acoustics and attempt to apply the techniques from PINNs to solve them.\n\nWEAKNESSES:\nThe paper itself feels less coherent, and seems like just an application of many existing PINN training techniques (e.g., hard constraint PINNs, collocation point sampling) into solving a certain problem, rather than providing a novel method or a coherent framework into solving a domain-specific problem.\n\nThe experimental section feels incomplete. Different point selection algorithms have not been extensively compared with, e.g., from that in Wu et. al. (2023). Furthermore, it would be interesting to see how the method can scale to more realistic acoustic problems (i.e., outside of 1D settings).\n\nThe paper itself also seems incomplete. The Appendix and the NeurIPS checklist are partially filled and have half-finished sentences.\n\nThe labels within the graphs can also be enlarged slightly to make them more readable.\n\nQUESTIONS:\n1. Can the optimal \\alpha be selected without having to test out different values of \\alpha? Is there some recommended value that can be used for different acoustics problems?\n\n2. Are there any relation between selecting points in high-amplitude regions with selecting points in high-residual regions, such as those methods benchmarked in Wu et. al. (2023)? In the sense that these two are indirectly doing the same thing, or they end up selecting very similar points anyway.\n\n3. In Figure 5, are there any intuition to why L1 loss peaks at larger \\alpha but L2 peaks at smaller \\alpha?\n\n4. How is the computation time of the methods proposed?\n\nLIMITATIONS:\nThe authors have provided limitations with selection of \\tau.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe manuscript treats the one dimensional wave equation with a PINN approach and discusses the imposition of boundary and initial conditions directly into the network, as common practice in PINNs. The authors then propose a quadrature scheme based on a coarse finite difference discretization of the wave equation.\n\nSTRENGTHS:\nThe imposition of the time derivative seems to be a novel construction. Furthermore, the construction seems not to be limited to the wave equation.\n\nWEAKNESSES:\nThe main weakness of the manuscript is the focus on the very special and simple toy problem of the one dimensional wave equation. Solving the one-dimensional wave equation with PINNs is only of academic interest and insights obtained from it for the training of PINNs might not generalize. More specifically:\n- The exact imposition of the time derivative should also work for general time dependent equations. The authors should comment on this.\n- The sampling strategy employing a finite difference simulation to determine regions of high sampling density is not a generalizable approach. If a finite difference solver for the equation at hand is available, a PINN solver is typically not required.\n- The authors determine an optimal function $\\tau$ via considering six concrete examples. There is no guarantee that this approach will generalize to different equation types and is therefore of limited practical use.\n- The authors might want to discuss the theoretical literature that proves the theoretical advantage of exactly imposed boundary conditions [1, 2, 3] and more elaborate constructions of distance functions.\n\n[1] https://proceedings.mlr.press/v190/muller22b/muller22b.pdf\n\n[2] https://arxiv.org/abs/2311.00529\n\n[3] https://www.sciencedirect.com/science/article/abs/pii/S0045782521006186\n\nQUESTIONS:\nSee weaknesses.\n\nLIMITATIONS:\nThe scope of the paper is too narrow.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-03"}

comments: SUMMARY:
This paper explores to solve the acoustic wave equation in the context of PINNs. Hard boundary and initial conditions are enforced by employing continuous functions within the PINN ansatz to ensure that these conditions are satisfied. A Dynamic Amplitude-Focused Sampling (DAFS) method is introduced to improve the efficiency of hard-constraint PINNs under a fixed number of sampling points.

STRENGTHS:
1. Propose a general hard constraint imposition formula which correctly imposes all boundary conditions and initial conditions as required.

WEAKNESSES:
1. Only the wave equation is discussed.
2. The proposed Dynamic Amplitude-Focused Sampling (DAFS) method is trivial.
3. There are no comparisons with other methods in the experiments.
4. In the experiments, the relative errors between exact solutions and predictions are not given.
5. In the context of PINNs, it is better to give explicitly the formulation of training loss. Training details are also lacking. 
6. Instead of tuning \tau (t) manually, it is better to train \tilde{u}(x,t) and \tau (t) simutanuously.
7. Many typos and grammar errors, such as "both and \alpha" in line 149, "x \in {\partial \Omega}_i" in line 125, "computational" in line 46.
8. The quality of Fig.7 should be improved.

QUESTIONS:
In line 41, what does the "basic function" mean? Is it the function \tau (t)?

LIMITATIONS:
Only the wave equation is discussed. There are no comparisons with other methods in the experiments.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2024-07-11

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper improves the training efficiency of original physics-informed neural networks to solve the 1D wave equation threefold: first by extending ansatz to also take the first derivative into account, second by a sampling method that focuses on high-amplitude regions, and third by a framework for domain decomposition.\n\nSTRENGTHS:\n+ The related work is well presented.\n+ The evaluation of the six candidate functions for \\tau in section 4.2 provides interesting insights. The authors explore an advanced selection method for \\tau based on the task at hand which might be an interesting research direction.\n\nWEAKNESSES:\n[Originality] While considering the first derivative for the ansatz is a good addition, the contribution is only minor. \nSampling more collocation points in the regions that might be more difficult to solve is a practical approach however the comparison and distinction to other sampling methods is missing.\nLastly if I understand the domain decomposition framework correctly, the contribution is to wrap the entire training into a loop and, based on the training process's results, increase or decrease the subdomain size. \n\nEvaluation results are only provided for the 1D wave equation. Further results for other differential equations are necessary to demonstrate the benefits of the proposed method.\n\n[Clarity] \nThe framework for domain decomposition is not presented clearly. While the flow chart in Figure 7 provides an overview of the method additional textual explanations in Section 4.4 are needed.\nThere were few to no remarks about the training regime (#training points, optimizer, learning rate…, etc.), making it more difficult to reproduce results.\nMinor remarks:\n-            N_pde is not introduced. It is probably the number of collocation points?\n-            Most of the Figures (e.g. Fig. 1, Fig 6.) are hard to read.\n-            Line 46: (…) optimal size of the computational [domain?] given (…)\n-            Line 149: Both [N_pde?] and alpha (…) \n-            Line 178: (…) In general, (...) performs better in general\n\nQUESTIONS:\nQ1: In the abstract the DAFS method is described as a method \"that optimizes the efficiency (…) under a FIXED number of sampling points. However in Section 3.2: \"This strategy optimally selects the number of points, N_pde, used in training.\" I assumed that DAFS only distributes the collection points into high and low-amplitude regions. Is that correct?\nQ2: How does DAFS compare to other state-of-the-art collocation point sampling methods? When should one use DAFS, and in which cases a different method?\n\nLIMITATIONS:\nWhile the authors clearly state that they are interested in the 1D wave equation it would have been interesting to see their proposed methods applied to the 2D wave equation of any other differential equations what are typically used in PINN benchmarks.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}