{
 "artifacts": null,
 "category": "training",
 "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_latex": [
  "$$I_k[w]=\\int_{\\mathbb{T}^{n}}e^{kH(Dw,x)}\\,dx,\\qquad \\int_{\\mathbb{T}^{n}}w\\,dx=0.$$",
  "$$\\frac{1}{k}\\log I_k[w]\\longrightarrow\\operatorname*{ess\\,sup}_{\\mathbb{T}^{n}}H(Dw,x).$$",
  "$$L_k(\\theta)=\\frac{1}{k}\\log\\left[\\frac{1}{N}\\sum_{i=1}^{N}\\exp\\left(k\\left(H(\\nabla_xu_\\theta(x_i),x_i)-c\\right)\\right)\\right],\\qquad \\tilde u_\\theta(x)=u_\\theta(x)-\\frac{1}{N}\\sum_{j=1}^{N}u_\\theta(x_j).$$",
  "$$H(Du_\\infty,x)\\le c,\\qquad \\mathcal{A}=\\{x:Du_\\infty(x)\\ \\text{exists and }H(Du_\\infty(x),x)=c\\}.$$"
 ],
 "id": 3013,
 "implementation": "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.",
 "math_summary": "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\u003e0 is an inverse-temperature parameter. The associated soft maximum satisfies (1/k)\\log I_k[w]\\to\\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(\\theta)=(1/k)\\log[(1/N)\\sum_i\\exp(k(H(\\nabla_xu_\\theta(x_i),x_i)-c))], with the network output centered to enforce the zero-mean constraint.",
 "math_tags": [
  "pde",
  "convex-analysis",
  "optimization"
 ],
 "ml_areas": [
  "loss",
  "training",
  "world-model"
 ],
 "paper": {
  "arxiv_id": "2609.01557",
  "arxiv_url": "https://arxiv.org/abs/2609.01557",
  "summary_what_math_gives_to_ml": "The paper provides a constructive zero-temperature variational principle for selecting weak KAM subsolutions: minimizing an exponential integral asymptotically minimizes the worst-case Hamilton–Jacobi violation. Its transferable asset is the combination of convex Hamiltonian structure, a globally normalized potential, and a theorem identifying a dynamically critical contact set. In neural PDE solvers and learned dynamical models, this suggests replacing average residual fitting with an annealed soft supremum and using long-time Lax–Oleinik near-contact scores for adaptive sampling. The approach is especially suitable for periodic value functions where worst-case constraint satisfaction and localization of invariant critical regions matter.",
  "title": "$L^\\infty$ Variational Approximation of the Aubry Set",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 3,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Soft-supremum weak-KAM loss",
 "url": "https://synthcore.org/idea/3013/soft-supremum-weak-kam-loss",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}