De Rham PINNs address topological failures in physics-informed neural networks
Original title: De Rham cohomological architectures for PINNs and PDEs with topological holes
Hodge-de Rham PINNs parameterize 1-forms to solve PDEs with topological holes and voids.
The authors introduce Hodge-de Rham PINNs to address failure modes of classical PINNs in scenarios with topological pathologies such as punctures, holes, poles, and voids. They parameterize 1-forms via the Hodge-Morrey-Friedrichs decomposition and draw a connection to de Rham cohomology through the HMF isomorphism theorem. PINN loss is performed on the parameterized 1-form through an associated PDE, remedying failures via a Sobolev-regular operator under Fréchet differentiation.
- Demonstrates methods on the Aharonov-Bohm effect
- Equips exhaustion functions when integrability condition not met on losses
- Provides theoretical guarantees on pathological safeguards
- Applies to scenarios of inexact forms and nontrivial harmonics
- Addresses closed-form operator relationships requiring integration
From the original article
1 min read at arxiv.orgWe 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.
By Andrew Gracyk · arxiv.org
A short excerpt; the article and its images belong to the publisher. Summary, key facts and assessment above are AI-written from the article and can be wrong.