# Caustic-Aware Hamiltonian Feedback Layer - ID: 1777 - Canonical URL: https://synthcore.org/idea/1777/caustic-aware-hamiltonian-feedback-layer - API JSON: https://synthcore.org/api/idea/1777.json - API Markdown: https://synthcore.org/api/idea/1777.md - Verification status: beats_baseline - Source: [arXiv:2608.04172](https://arxiv.org/abs/2608.04172) - Category: dynamics - Solves: stability, accuracy - ML areas: optimizer, training-dynamics, rl, world-model - Math tags: control-theory, dynamical-systems, differential-geometry, geometry - Ratings: usefulness 8/10; difficulty 6/10; novelty 7/10 ## Idea description Represent a neural controller as the projection of a learned Hamiltonian stable manifold rather than learning a state-to-action map without geometric constraints. Train a manifold chart together with its invariance equation, and reject or branch-switch near points where the manifold projection becomes singular. The resulting controller exposes a measurable boundary between single-valued smooth feedback and multivalued or hysteretic feedback. ## Mathematical statement For a controlled state x in R^n with dynamics dot{x}=f(x,u) and running cost ell(x,u), define the Hamiltonian H(x,p)=max_{u in U}[p^T f(x,u)-ell(x,u)]. The characteristic equations are dot{x}=f(x,mu(x,p)) and dot{p}=nabla_x ell(x,mu)-[D_x f(x,mu)]^T p, where p is the costate and mu(x,p)=argmax_u[p^T f(x,u)-ell(x,u)]. The feedback manifold is the stable set Lambda=W^s(N) of a target invariant manifold N in phase space. A smooth feedback exists locally only when the projection pi:Lambda to x is locally invertible. If a chart z maps to (x(z),p(z)), its caustic indicator is sigma_min(D_z x); a caustic occurs when sigma_min(D_z x)=0. The neural adaptation learns a chart (x_phi(z),p_phi(z)) using the Hamiltonian invariance residual and penalizes small projection singular values. ## Key formulas - $$\mu(x,p)=\arg\max_{u\in\mathcal U}\left[p^{\mathsf T}f(x,u)-\ell(x,u)\right],\qquad \dot{x}=f(x,\mu(x,p)),\qquad \dot{p}=\nabla_x\ell(x,\mu)-[D_xf(x,\mu)]^{\mathsf T}p.$$ - $$\Lambda=W^s(N)=\left\{(x,p):\lim_{t\to\infty}\varphi_t(x,p)\in N\right\},\qquad \text{smooth feedback requires }\operatorname{rank}D_zx(z)=n.$$ - $$\mathcal L_{\mathrm{inv}}(\phi)=\mathbb E_z\left\|D_zp_\phi(z)\,\dot z-\nabla_x\ell(x_\phi(z),u_\phi(z))+[D_xf(x_\phi(z),u_\phi(z))]^{\mathsf T}p_\phi(z)\right\|^2,$$ - $$\mathcal L_{\mathrm{cau}}(\phi)=\mathbb E_z\left[\max\left(0,\tau-\sigma_{\min}(D_zx_\phi(z))\right)^2\right],\qquad u_\phi(z)=\mu(x_\phi(z),p_\phi(z)).$$ ## Implementation notes Use this first as a controlled neural ODE layer or as a parameter-space optimizer. Let x be the plant state and let an MLP parameterize a phase-space chart (x_phi(z),p_phi(z)). Compute the action with the analytic maximizer u=mu(x,p) whenever possible; otherwise use a differentiable inner optimization over u. For each minibatch, sample chart coordinates z, evaluate x and p, compute u, use automatic differentiation to obtain D_z x, D_z p, and D_x f, and add the invariance residual from the displayed equation. Add a terminal-attraction loss that pulls integrated trajectories toward the target manifold N, plus the caustic barrier L_cau. The paper supplies the Hamiltonian equations, stable-manifold construction, isotropic-manifold result, and projection criterion; the network estimates the manifold chart and its Jacobians. The first cheap experiment should reproduce the scalar system dot{x}=x-4u^3+6u with ell=3u^2(1-u^2), for which H=px-p^4+3p^2, u=p, and Lambda is x=p^3-3p. Train a one-dimensional MLP chart and compare it with a direct MLP controller and policy-gradient control. The predicted signature is projection degeneracy at p=+/-1, since dx/dp=3p^2-3=0, giving caustic state locations x=+/-2. For -2