{
 "artifacts": null,
 "category": "geometry",
 "description": "Use the discrepancy between a learned potential and its long-time backward Lax–Oleinik evolution to identify dynamically critical states. Persistent near-contact points are candidates for the Aubry set and can guide adaptive collocation, while states with large gaps can receive fewer training samples.",
 "formulas_latex": [
  "$$\\mathcal{A}=\\{x\\in\\mathbb{T}^{n}:u_{\\infty}(x)=u_{-}(x)\\}=\\{x\\in\\mathbb{T}^{n}:Du_{\\infty}(x)\\ \\text{exists and }H(Du_{\\infty}(x),x)=c\\}.$$",
  "$$L(v,x)=\\sup_{p\\in\\mathbb{R}^{n}}\\{p\\cdot v-H(p,x)\\}.$$",
  "$$(T_t^-f)(x)=\\inf_{\\substack{\\gamma:[0,t]\\to\\mathbb{T}^n\\\\\\gamma(t)=x}}\\left[f(\\gamma(0))+\\int_0^tL(\\dot\\gamma(s),\\gamma(s))\\,ds\\right].$$",
  "$$g_t(x)=u_\\theta(x)-(T_t^-u_\\theta)(x)+ct,\\qquad \\mathcal{A}_{\\mathrm{probe}}(\\varepsilon)=\\{x:|g_t(x)|\\le\\varepsilon\\}.$$"
 ],
 "id": 3014,
 "implementation": "Attach this module between neural PDE training iterations, after the network has produced a preliminary value function. Evaluate u_theta on a dense periodic candidate grid and approximate T_t^-u_theta at each endpoint. For a first MVP, discretize time into M steps of size Delta t and restrict velocities to a finite stencil V. Initialize A_0(y)=u_theta(y), then apply the dynamic-programming update A_{r+1}(x)=min_{v in V}[A_r(x-Delta t v)+Delta t L(v,x-Delta t v)], using periodic indexing. Set A_M(x) as the approximation to T_t^-u_theta(x), compute g_t(x)=u_theta(x)-A_M(x)+ct, and subtract its median to remove numerical additive drift. Define contact weights q_i=exp(-|g_t(x_i)|/tau). Use a 50/50 mixture of uniform collocation points and points sampled proportional to q_i; alternatively, add an auxiliary penalty mean_i min(|g_t(x_i)|,epsilon)^2 only after the base Hamiltonian loss is stable. The paper supplies the contact-set characterization, while the velocity stencil, finite horizon, grid interpolation, and tolerance are empirical approximations. Test first on a 1D or 2D periodic mechanical Hamiltonian with a reference Aubry set from a high-resolution solver. Compare uniform sampling with contact-guided sampling at equal H evaluations. Measure Aubry-set Hausdorff distance, dense-grid Hamiltonian violation, and value error. Success is faster error reduction near separatrices and improved contact-set localization at equal sample count.",
 "math_summary": "The paper characterizes the projected Aubry set through the limiting potential u_\\infty and its associated backward weak KAM solution u_-: \\mathcal{A}=\\{x:u_\\infty(x)=u_-(x)\\}, equivalently \\mathcal{A}=\\{x:Du_\\infty(x)\\text{ exists and }H(Du_\\infty(x),x)=c\\}. For a Tonelli Hamiltonian H, its Legendre transform is L(v,x)=\\sup_{p\\in\\mathbb{R}^n}\\{p\\cdot v-H(p,x)\\}, where v is velocity and p is momentum. The backward Lax–Oleinik operator is (T_t^-f)(x)=\\inf_{\\gamma(t)=x}[f(\\gamma(0))+\\int_0^tL(\\dot\\gamma(s),\\gamma(s))ds]. A backward weak KAM solution is invariant under this evolution up to the critical normalization, T_t^-u_-=u_-+ct. The neural diagnostic is therefore g_t(x)=u_\\theta(x)-(T_t^-u_\\theta)(x)+ct; small persistent absolute gap indicates a candidate contact or Aubry point.",
 "math_tags": [
  "dynamical-systems",
  "geometry",
  "pde",
  "convex-analysis",
  "optimization"
 ],
 "ml_areas": [
  "world-model",
  "rl",
  "training",
  "data-augmentation"
 ],
 "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": 6,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "generalization"
 ],
 "title": "Lax–Oleinik Aubry contact probe",
 "url": "https://synthcore.org/idea/3014/lax-oleinik-aubry-contact-probe",
 "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
   }
  }
 }
}