Estimatable variation neural networks and their application to scalar hyperbolic conservation laws

TL;DR

This work introduces estimatable variation neural networks (EVNNs), a framework that enforces computable bounds on a BV-type norm via the space $BMV$, enabling robust approximation of non-smooth functions arising in scalar hyperbolic conservation laws. The authors prove a universal approximation property for EVNNs within $BMV$ and establish convergence and existence results for minimizers in both ODEs and conservation laws when the target solution lies in $BMV$, supported by explicit, computable loss sequences. They also develop a practical initialization and training strategy and demonstrate the approach on Burgers’ equation, including standing/moving shocks, rarefaction, and a sine test, highlighting stability and the role of entropy admissibility. The results provide a principled connection between function-variation control and neural-network-based solution learning for non-smooth PDEs, offering a rigorous, adaptable tool for data-constrained forward and inverse problems with shocks.

Abstract

We introduce estimatable variation neural networks (EVNNs), a class of neural networks that allow a computationally cheap estimate on the $BV$ norm motivated by the space $BMV$ of functions with bounded M-variation. We prove a universal approximation theorem for EVNNs and discuss possible implementations. We construct sequences of loss functionals for ODEs and scalar hyperbolic conservation laws for which a vanishing loss leads to convergence. Moreover, we show the existence of sequences of loss minimizing neural networks if the solution is an element of $BMV$. Several numerical test cases illustrate that it is possible to use standard techniques to minimize these loss functionals for EVNNs.

Figures (10)

• Figure 1: Possible implementations for the proposed architecture for EVNNs from $\mathds{R}^d$ to $\mathds{R}$. It is straightforward to generalize this architecture to mappings from $\mathds{R}^d$ to $\mathds{R}^N$.
• Figure 2: The first network initialization trained for the sine initial data test case with periodic boundary condition.
• Figure 3: Illustration of Example \ref{['App:ex:EMVar']}.
• Figure 4: Learning exponential decay by enforcing the ODE. The reference solution as well as the average and range at each point in time of 100 network runs are plotted. We show the network solution with $\mathcal{E}=0.1$.
• Figure 5: Poisson problem in one dimension with non-smooth solution. The reference solution as well as the average and range at each point in time of 100 network runs are plotted.

Theorems & Definitions (59)

• Theorem 1.1: Informal
• Definition 2.1: $BMV$
• Lemma 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Lemma 2.7
• proof