Geometric and Physical Constraints Synergistically Improve Neural PDE Integration
Keywords: Geometric deep learning, physics-constrained neural networks, PDE integration
TL;DR: Symmetry constraints and conservation laws give non-redundant improvements to neural PDE integration
Abstract: Neural PDE surrogates can improve on cost-accuracy tradeoffs of classical solvers, but often generalize poorly to new initial conditions, accumulate errors over time. To close the performance gap between training and long-term inference, we constrain neural surrogates with symmetry equivariance and physical conservation laws as hard constraints, using novel input and output layers that support scalar and vector fields on the staggered grids commonly used in computational fluid dynamics. We systematically investigate how these constraints affect accuracy, individually and in combination, on two challenging tasks: shallow water equations with closed boundaries and decaying incompressible turbulence. Compared to a strong baseline, both types of constraints improve performance consistently across autoregressive prediction steps, accuracy measures, and network sizes. Symmetries are more effective but do not make physical constraints redundant. Doubly-constrained surrogates were more accurate for the same network and dataset sizes, and generalized better to initial conditions and durations beyond the range of training data.
Primary Area: learning on time series and dynamical systems
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.
Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.
Reciprocal Reviewing: I understand the reciprocal reviewing requirement as described on https://iclr.cc/Conferences/2025/CallForPapers. If none of the authors are registered as a reviewer, it may result in a desk rejection at the discretion of the program chairs. To request an exception, please complete this form at https://forms.gle/Huojr6VjkFxiQsUp6.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Submission Number: 13848
Loading