From Numerical Simulators of PDEs to Neural Emulators and Back
Mirrored from arXiv — Machine Learning for archival readability. Support the source by reading on the original site.
Computer Science > Machine Learning
Title:From Numerical Simulators of PDEs to Neural Emulators and Back
Abstract:Simulation is central to modern engineering and science, but the cost of numerical solvers for partial differential equations (PDEs) remains a bottleneck whenever fast or many-query evaluations are required. Neural emulators trained on solver-generated data promise significant speedups, yet they are usually framed as opaque alternatives to the very methods that produce their training signal. This thesis argues the two paradigms are more alike than different: neural architectures mirror classical discretizations, their errors are amenable to the same spectral analysis, and insight flows profitably in both directions. We approach the relationship by disentangling the multiple roles a solver plays in the emulator learning pipeline. Mode-wise Fourier analysis then provides a common language in which solver errors, architectural inductive biases, and training objectives can all be read off simultaneously. Taken together, this allows synthesizing three contributions. (1) APEBench, a comprehensive benchmarking suite for autoregressive neural emulators of PDEs that uses fast differentiable pseudo-spectral solvers in JAX. (2) Progressively Refined Differentiable Physics, an investigation of the effect of unconverged solvers on surrogate training. (3) Neural Emulator Superiority, an analysis of the influence of numerical errors and architectural inductive biases.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2608.24547 [cs.LG] |
| (or arXiv:2608.24547v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.24547
arXiv-issued DOI via DataCite (pending registration)
|
Access Paper:
- View PDF
- HTML (experimental)
- TeX Source
References & Citations
Bookmark
Bibliographic and Citation Tools
Code, Data and Media Associated with this Article
Demos
Recommenders and Search Tools
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
More from arXiv — Machine Learning
-
SLM-Conditioned Hierarchical Relation Routing for Labeled Property Graph Learning
Aug 28
-
NeuronFuzz: Safety Neuron Guided Fuzzing for LLM Safety Evaluation
Aug 28
-
Pruning Binarized Neural Networks: A Dedicated Framework and Globally Weighted Algorithms
Aug 28
-
Muon with Finite Newton-Schulz: The Smoothing Benefit in Nonsmooth Nonconvex Optimization
Aug 28
Discussion (0)
Sign in to join the discussion. Free account, 30 seconds — email code or GitHub.
Sign in →No comments yet. Sign in and be the first to say something.