Physics-Informed Residual Flows

Jephte Abijuru, Mayank Nagda, Phil Ostheimer, Sebastian Josef Vollmer, Marius Kloft, Sophie Fellenz
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:185-210, 2026.

Abstract

Physics-Informed Neural Networks (PINNs) embed physical laws into deep learning models. However, conventional PINNs often suffer from failure modes leading to inaccurate solutions. We trace these failure modes to two structural pathologies: gradient shattering, where gradients degrade with depth and provide little training signal, and flow mismatch, where training pushes predictions along trajectories that diverge from the PDE solution path. We introduce ResPINNs, which reformulate PINNs as residual flows, networks that iteratively refine their own predictions through explicit corrective steps, in the spirit of classical iterative solvers. Our analysis shows that this design mitigates both pathologies by keeping updates aligned with descent and by preserving informative gradients across depth. Extensive experiments on PDE benchmarks confirm that ResPINNs achieve higher accuracy with substantially fewer parameters than conventional architectures.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-abijuru26a, title = {Physics-Informed Residual Flows}, author = {Abijuru, Jephte and Nagda, Mayank and Ostheimer, Phil and Vollmer, Sebastian Josef and Kloft, Marius and Fellenz, Sophie}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {185--210}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/abijuru26a/abijuru26a.pdf}, url = {https://proceedings.mlr.press/v306/abijuru26a.html}, abstract = {Physics-Informed Neural Networks (PINNs) embed physical laws into deep learning models. However, conventional PINNs often suffer from failure modes leading to inaccurate solutions. We trace these failure modes to two structural pathologies: gradient shattering, where gradients degrade with depth and provide little training signal, and flow mismatch, where training pushes predictions along trajectories that diverge from the PDE solution path. We introduce ResPINNs, which reformulate PINNs as residual flows, networks that iteratively refine their own predictions through explicit corrective steps, in the spirit of classical iterative solvers. Our analysis shows that this design mitigates both pathologies by keeping updates aligned with descent and by preserving informative gradients across depth. Extensive experiments on PDE benchmarks confirm that ResPINNs achieve higher accuracy with substantially fewer parameters than conventional architectures.} }
Endnote
%0 Conference Paper %T Physics-Informed Residual Flows %A Jephte Abijuru %A Mayank Nagda %A Phil Ostheimer %A Sebastian Josef Vollmer %A Marius Kloft %A Sophie Fellenz %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-abijuru26a %I PMLR %P 185--210 %U https://proceedings.mlr.press/v306/abijuru26a.html %V 306 %X Physics-Informed Neural Networks (PINNs) embed physical laws into deep learning models. However, conventional PINNs often suffer from failure modes leading to inaccurate solutions. We trace these failure modes to two structural pathologies: gradient shattering, where gradients degrade with depth and provide little training signal, and flow mismatch, where training pushes predictions along trajectories that diverge from the PDE solution path. We introduce ResPINNs, which reformulate PINNs as residual flows, networks that iteratively refine their own predictions through explicit corrective steps, in the spirit of classical iterative solvers. Our analysis shows that this design mitigates both pathologies by keeping updates aligned with descent and by preserving informative gradients across depth. Extensive experiments on PDE benchmarks confirm that ResPINNs achieve higher accuracy with substantially fewer parameters than conventional architectures.
APA
Abijuru, J., Nagda, M., Ostheimer, P., Vollmer, S.J., Kloft, M. & Fellenz, S.. (2026). Physics-Informed Residual Flows. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:185-210 Available from https://proceedings.mlr.press/v306/abijuru26a.html.

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