# Soft-supremum weak-KAM loss

- ID: 3013
- Canonical URL: https://synthcore.org/idea/3013/soft-supremum-weak-kam-loss
- API JSON: https://synthcore.org/api/idea/3013.json
- API Markdown: https://synthcore.org/api/idea/3013.md
- Verification status: unverified
- Source: [arXiv:2609.01557](https://arxiv.org/abs/2609.01557)
- Category: training
- Solves: stability, accuracy, sample-efficiency
- ML areas: loss, training, world-model
- Math tags: pde, convex-analysis, optimization
- Ratings: usefulness 6/10; difficulty 3/10; novelty 5/10

## 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.

## 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.

## Key formulas

- $$I_k[w]=\int_{\mathbb{T}^{n}}e^{kH(Dw,x)}\,dx,\qquad \int_{\mathbb{T}^{n}}w\,dx=0.$$
- $$\frac{1}{k}\log I_k[w]\longrightarrow\operatorname*{ess\,sup}_{\mathbb{T}^{n}}H(Dw,x).$$
- $$L_k(\theta)=\frac{1}{k}\log\left[\frac{1}{N}\sum_{i=1}^{N}\exp\left(k\left(H(\nabla_xu_\theta(x_i),x_i)-c\right)\right)\right],\qquad \tilde u_\theta(x)=u_\theta(x)-\frac{1}{N}\sum_{j=1}^{N}u_\theta(x_j).$$
- $$H(Du_\infty,x)\le c,\qquad \mathcal{A}=\{x:Du_\infty(x)\ \text{exists and }H(Du_\infty(x),x)=c\}.$$

## 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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.