Soft-supremum weak-KAM loss
Implementation & benchmark of arXiv:2609.01557 — $L^\infty$ Variational Approximation of the Aubry Set
Source paper: $L^\infty$ Variational Approximation of the Aubry Set arXiv:2609.01557 ⓘ · analyzed Sep 2, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Train a neural periodic potential to minimize an exponential variational functional rather than a mean-squared Hamilton–Jacobi residual. Increasing the inverse-temperature parameter concentrates optimization on the worst violating locations, encouraging a learned critical subsolution whose equality set represents dynamically important regions.
Formulas
Mathematical statement
The paper minimizes I_k[w]=\int_{\mathbb{T}^{n}}e^{kH(Dw,x)}dx subject to \int_{\mathbb{T}^{n}}w\,dx=0. Here x\in\mathbb{T}^n is the periodic state, w is a scalar potential, Dw is its spatial gradient, H(p,x) is a smooth periodic Tonelli Hamiltonian, p is momentum, and k>0 is an inverse-temperature parameter. The associated soft maximum satisfies (1/k)\log I_k[w] o\operatorname*{ess\,sup}_{x\in\mathbb{T}^n}H(Dw,x). For critical value c, the limiting minimizer is a critical subsolution with H(Du_\infty,x)\le c. Under the paper's Mather-quotient assumption, equality with c occurs precisely on the projected Aubry set. The neural adaptation uses the stable empirical objective L_k( heta)=(1/k)\log[(1/N)\sum_i\exp(k(H(
abla_xu_ heta(x_i),x_i)-c))], with the network output centered to enforce the zero-mean constraint.
Implementation notes
Integrate this at the loss level of a coordinate-based neural PDE solver. Let u_theta map periodic coordinates x in T^n to a scalar, and use automatic differentiation to compute grad_x u_theta at uniformly sampled collocation points x_i. Center the output each batch as tilde_u_theta(x_i)=u_theta(x_i)-mean_j u_theta(x_j), satisfying the paper's normalization. Compute h_i=H(grad_x tilde_u_theta(x_i),x_i)-c. For numerical stability, set z_i=k h_i, m=max_i z_i, and evaluate L_k=m/k+log(mean_i exp(z_i-m))/k. Backpropagate this loss with Adam. Start at k=1 or 2 and increase k geometrically only after the batch maximum violation is sufficiently small; clip k to prevent gradients from collapsing onto one point. The mathematical object computed from the paper is the exponential soft supremum; the integral is estimated by Monte Carlo, while c is either known or estimated separately. The first experiment should use a 2D periodic mechanical Hamiltonian H(p,x)=0.5||p||^2+V(x), with a finite-difference reference solution. Compare this loss against mean-squared residual and a hinge loss at equal gradient-evaluation and collocation budgets. Measure dense-grid maximum H-c, the fraction of violating points, value-function error, and seed variance. Success means lower worst-case violation and a more stable equality set without more than 20 percent extra wall-clock time.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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