Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations

TL;DR

The paper tackles solving fully nonlinear Hamilton–Jacobi equations in high dimensions by grounding neural-network training in residual minimization of monotone finite-difference discretizations, ensuring convergence to the viscosity solution via Barles–Souganidis theory. A residual operator ${\mathcal{R}}$ and loss $L(u)=\frac{1}{q}\|{\mathcal{R}}(u)\|_q^q$ are defined on a geometric graph, with a monotone numerical Hamiltonian ${\mathcal{H}}$, and the authors prove the unique critical point of $L$ corresponds to the discrete solution under conditions (H1)-(H3). The framework extends to time-dependent problems with implicit schemes showing no CFL restriction for solvability, and a spectral analysis reveals the residual Jacobian’s condition number scales as $O(h^{-1})$, motivating multi-level warm-start training. Numerical experiments in 1D and high dimensions, including obstacle problems and Hamilton-Jacobi-Isaacs equations, demonstrate scalability, robustness, and the potential to apply Level Set Methods in high dimensions without dense meshes. This residual-minimization approach thus provides a principled, scalable path to solving high-dimensional HJ equations with practical impact on level-set methods and high-dimensional interface problems.

Abstract

We derive sufficient conditions under which residual minimization yields well-posed discrete solutions for nonlinear equations defined by monotone finite--difference discretizations. Our analysis is motivated by the challenge of solving fully nonlinear Hamilton--Jacobi (HJ) equations in high dimensions by means of a Neural Network, which is trained by minimizing residuals arising from monotone discretizations of the Hamiltonian. While classical theory ensures that consistency and monotonicity imply convergence to the viscosity solution, treating these discrete systems as optimization problems introduces new analytical hurdles: solvability and the uniqueness of local minima do not follow from monotonicity alone. By establishing the well--posedness of these optimization--based solvers, our framework enables the adaptation of Level Set Methods to high--dimensional settings, unlocking new capabilities in applications such as high--dimensional segmentation and interface tracking. Finally, we observe that these arguments extend almost directly to degenerate elliptic or parabolic PDEs on graphs equipped with monotone graph Laplacians.

Figures (5)

• Figure 1: Evolution of the loss function $L(u):= \|{\mathcal{R}}(u)\|_2^2$ with different choices of $\Delta x$, from $0.05$ to $0.00625$. Full gradient descent is implemented in the space of grid functions $\mathbb{R}^{N+1}$, with $N = 1/\Delta x$. On the left, we see the evolution of the loss on a fixed grid, i.e. with constant $\Delta x$. On the right, we see the loss evolution on a varying grid with decreasing values of $\Delta x$. Every time the grid is refined, the grid function from the previous step is interpolated in the new grid by taking constant values between the former grid points.
• Figure 2: On the left, we see the solution $u$ associated to the experiments in Figure \ref{['fig:gradient descent']} (left), for different values of $\Delta x$. On the center and right plots, for the last case (with $\Delta x = 0.00625$), we see (in blue) the residual function ${\mathcal{R}}(u)$ after $10^5$ iterations of gradient descent; and (in orange) the eigenvectors of the Jacobian matrix $D{\mathcal{R}}(u)$ associated to the smallest eigenvalue (of size $0.079$).
• Figure 3: Summary of the experiments described in section \ref{['subsec: experiment 1 NN']}, for the 1D eikonal equation. Each experiment was performed $100$ times independently. In the last experiment (in red), we used a training schedule as in Algorithm 2, with $\Delta x$ taking the values $1/20, 1/100, 1/500$ and $1/1000$ and $\lambda$ taking the values $2, 1, 0.5$ and $0.1$. The left plot represents the evolution of the $L^\infty$-error in average over the 100 repetitions. The central plot represents the $L^\infty$-error at the end of the training procedure. On the right, we see the number of iterations before the stopping criterion was reached. In the two latter plots, points represent the mean over $100$ i.i.d. experiments, and the intervals are delimited by the standard deviation.
• Figure 4: The zero-level set of the solution at time $t=0$, $t=1$ and $t=2$. The figure represents a central 2D slice of the 5D domain, rotated so that the drift direction $a_0 = (1, 1, \ldots, 1)$ corresponds to the positive direction of the $x$-axis.
• Figure 5: Above: DNN scheme, Below: reference solution

Theorems & Definitions (14)

• Lemma 1
• proof
• Theorem 1: Well-posedness of residual minimization
• proof
• Theorem 2: Stability
• proof
• Proposition 1
• proof
• Theorem 3: Critical points - time dependent case
• proof