De Rham cohomological architectures for PINNs and PDEs with topological holes
Abstract
We attempt to alleviate failure modes of classical physics-informed neural networks (PINNs) in scenarios of topological pathologies corresponding to punctures, holes, poles, and voids by introducing de Rham cohomological augmentations, or Hodge-de Rham PINNs. Vanilla PINNs often encounter pitfalls in hole-based settings due to discontinuous interfaces, nonzero closed path integration, path dependence, correspondence to exact forms via the exterior derivative, etc., therefore we absolve traditional failures via parameterizing a 1-form by the Hodge-Morrey-Friedrichs (HMF) decomposition for manifolds with boundary, drawing a connection to de Rham cohomology such as through the HMF isomorphism theorem. 1-forms can act as representations of scalar-valued PDE solutions through integral operator mappings. We propose performing a PINN-based loss on the parameterized 1-form through an associated PDE, which subsequently solves the potential-based PINN problem and moreover remedies the failures of vanilla PINNs via the Sobolev-regular operator under a Fréchet differentiation interconnection. The operator relationship often has a nontrivial closed-form and often requires integration or a case-by-case analysis, which we address thoroughly in our experiments. When an integrability condition is not met on our losses, we can equip exhaustion functions to train the networks. We demonstrate our methods on the Aharonov-Bohm effect, which, to our understanding, is largely nonexistent in PINN literature. We develop theoretical guarantees of our methods and demonstrate our pathological safeguards on a broad selection of topologically-adverse problems, including scenarios of inexact forms, and those of nontrivial harmonics and coexact terms.
Related
- A support preserving homotopy for the de Rham complex with boundary decay estimates
- Universal norms and de Rham representations
- A Counterexample to Cohomological Rigidity of Toric Manifolds
- Revisiting Natural Gradient Descent for PINNs Using Adaptive Collocation Strategies
- A question of Ulam and a theorem of de Rham
BibTeX
@misc{gracyk2026rham,
title = {{De Rham cohomological architectures for PINNs and PDEs with topological holes}},
author = {Andrew Gracyk},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05276},
eprint = {2611.05276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05276},
}
RIS
TY - UNPB
TI - De Rham cohomological architectures for PINNs and PDEs with topological holes
AU - Andrew Gracyk
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05276
UR - https://arxiv.org/abs/2611.05276
AB - We attempt to alleviate failure modes of classical physics-informed neural networks (PINNs) in scenarios of topological pathologies corresponding to punctures, holes, poles, and voids by introducing de Rham cohomological augmentations, or Hodge-de Rham PINNs. Vanilla PINNs often encounter pitfalls in hole-based settings due to discontinuous interfaces, nonzero closed path integration, path dependence, correspondence to exact forms via the exterior derivative, etc., therefore we absolve traditional failures via parameterizing a 1-form by the Hodge-Morrey-Friedrichs (HMF) decomposition for manifolds with boundary, drawing a connection to de Rham cohomology such as through the HMF isomorphism theorem. 1-forms can act as representations of scalar-valued PDE solutions through integral operator mappings. We propose performing a PINN-based loss on the parameterized 1-form through an associated PDE, which subsequently solves the potential-based PINN problem and moreover remedies the failures of vanilla PINNs via the Sobolev-regular operator under a Fréchet differentiation interconnection. The operator relationship often has a nontrivial closed-form and often requires integration or a case-by-case analysis, which we address thoroughly in our experiments. When an integrability condition is not met on our losses, we can equip exhaustion functions to train the networks. We demonstrate our methods on the Aharonov-Bohm effect, which, to our understanding, is largely nonexistent in PINN literature. We develop theoretical guarantees of our methods and demonstrate our pathological safeguards on a broad selection of topologically-adverse problems, including scenarios of inexact forms, and those of nontrivial harmonics and coexact terms.
ER -