Native neural network classes for solving partial differential equations
FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equations
Machine Learning
Summary
Solving partial differential equations (PDEs) involving vector fields like fluid flow or electromagnetic fields usually requires special mathematical spaces. The authors introduce new neural network classes that naturally fit these spaces without needing complex meshes. These classes use special potentials to represent fields and handle jumps or discontinuities in the solutions. This approach leads to a new method for solving PDEs that works well even with tricky data like discontinuous coefficients or shocks.
What this means in practice
- •For computational engineers: Solve elliptic and curl-curl PDEs with discontinuous coefficients using a neural method designed for native function spaces and natural weak formulations.
- •For numerical analysts: Model linear transport equations and conservation laws with shocks through neural networks that handle fluxes with discontinuities.
Authors
Shun Zhang
Abstract
We construct neural approximation classes native to the graph spaces H(div) and H(curl), in two and three dimensions and, for H(div), in any dimension. Every realization lies in the space for all parameter values, and with kinked potentials, such as ReLU networks, the admissible jumps appear at finite width. The classes are images of scalar and componentwise networks under fixed operators of the de Rham complex, and do not involve a mesh or finite element emulation. For H(div) in R^n two native classes are given on an equal footing, with a skew-symmetric potential $A$: $\mathrm{Div}\,A+R_nq+\mathbf{h}$, with the divergence $q$ as an explicit unknown, and $\mathrm{Div}\,A+\mathbf{z}$ with an $H^1$ field $\mathbf{z}$; for H(curl) the analogous classes are $\mathrm{grad}\,φ+Sr+\mathbf{h}$ in two dimensions and $\mathrm{grad}\,φ+\mathbf{z}$ in two and three dimensions. In all of them every interface jump of the field is carried by the potential term, $\mathrm{Div}\,A$ or $\mathrm{grad}\,φ$, while the remaining part has no interface jump (it is an $H^1$ field in the regular-decomposition classes); the classes with $\mathbf{z}$ are the componentwise approach enriched by this term. Known or learned interface geometry enters the potential through factors with trainable amplitudes, and the remaining part if the divergence jumps. The classes lead to the FOSLS-deRhaNN method, first-order system least squares with de Rham neural networks, whose loss is the least-squares functional posed in the natural spaces of the weak formulation; for elliptic equations this includes $H^{-1}$ right-hand sides and $H^{1/2}$ Dirichlet data. Elliptic equations with discontinuous coefficients and curl-curl problems are treated as instances, with the functional equivalent to the error; linear transport with discontinuous solutions and conservation laws with shocks use the same flux classes.