Computing high-order mixed derivatives in physics-informed neural networks using multi-index Bell polynomials
Abstract
Physics-informed neural networks for high-order partial differential equations require mixed input derivatives and their gradients with respect to the network parameters. Standard implementations obtain an order-K derivative by repeated automatic differentiation. We instead organize the forward recursion of the multivariate Faà di Bruno formula and its explicit backpropagation over a prescribed downward-closed set of multi-indices, with the Bell-polynomial convolutions tabulated once. The forward pass carries only the derivatives the differential operator requires. The backward pass propagates gradients from losses formed from any subset of them, including nonlinear products and coupled fields. Both recursions are exact up to roundoff and avoid nested computational graphs. An independent Taylor-jet implementation, symbolic checks of the test problems, and finite differences verify derivatives and loss gradients through order seven. On one CPU core, the method evaluates 330 mixed derivatives with respect to four inputs through order seven and the corresponding loss gradient without the memory failures observed for several nested implementations. Numerical tests include third-, fifth-, and seventh-order dispersive equations, incompressible flow, and a manufactured five-field electrohydrodynamic system. The seventh-order Zakharov-Kuznetsov test in 3+1 dimensions has a relative solution error of 6×10−4.
Related
- Beyond white- and black-box modeling tools in optical communications and optical computing: physics-informed data-driven modeling
- Accelerating noisy variational quantum algorithms with physics-informed denoising networks
- Physics-informed neural networks for soliton solutions and parameter identification of the two-mode Korteweg–de Vries equation
- Lazy training of quantum physics informed neural networks
- Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications
BibTeX
@misc{imoto2026computing,
title = {{Computing high-order mixed derivatives in physics-informed neural networks using multi-index Bell polynomials}},
author = {Fumihiro Imoto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03768},
eprint = {2609.03768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03768},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Computing high-order mixed derivatives in physics-informed neural networks using multi-index Bell polynomials
AU - Fumihiro Imoto
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03768
UR - https://arxiv.org/abs/2609.03768
AB - Physics-informed neural networks for high-order partial differential equations require mixed input derivatives and their gradients with respect to the network parameters. Standard implementations obtain an order-K derivative by repeated automatic differentiation. We instead organize the forward recursion of the multivariate Faà di Bruno formula and its explicit backpropagation over a prescribed downward-closed set of multi-indices, with the Bell-polynomial convolutions tabulated once. The forward pass carries only the derivatives the differential operator requires. The backward pass propagates gradients from losses formed from any subset of them, including nonlinear products and coupled fields. Both recursions are exact up to roundoff and avoid nested computational graphs. An independent Taylor-jet implementation, symbolic checks of the test problems, and finite differences verify derivatives and loss gradients through order seven. On one CPU core, the method evaluates 330 mixed derivatives with respect to four inputs through order seven and the corresponding loss gradient without the memory failures observed for several nested implementations. Numerical tests include third-, fifth-, and seventh-order dispersive equations, incompressible flow, and a manufactured five-field electrohydrodynamic system. The seventh-order Zakharov-Kuznetsov test in 3+1 dimensions has a relative solution error of 6×10^−4.
ER -