# Jacobian-Frozen Stable Rollouts

- ID: 3098
- Canonical URL: https://synthcore.org/idea/3098/jacobian-frozen-stable-rollouts
- API JSON: https://synthcore.org/api/idea/3098.json
- API Markdown: https://synthcore.org/api/idea/3098.md
- Verification status: unverified
- Source: [arXiv:2609.03348](https://arxiv.org/abs/2609.03348)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: world-model, rnn, ssm, training-dynamics, regularization
- Math tags: dynamical-systems, control-theory, spectral-theory, linear-algebra, approximation-theory
- Ratings: usefulness 7/10; difficulty 5/10; novelty 5/10

## Idea description

Replace an expensive nonlinear rollout of a recurrent or neural state-space model by a locally affine rollout whose Jacobian is evaluated once at the current state and then frozen over a short horizon. Use the resulting transition matrix as an explicit stability monitor and optionally penalize or clip its spectral radius, reducing exploding long-horizon predictions without forcing the entire nonlinear network to be globally contractive.

## Mathematical statement

Let $g_{\theta}(x,u)$ be a learned one-step state transition, with state $x\in\mathbb{R}^{n}$, input $u\in\mathbb{R}^{m}$, and parameters $\theta$. At the current operating point $(x_k,u_k)$, define $A_k=\partial g_{\theta}/\partial x|_{(x_k,u_k)}$, $B_k=\partial g_{\theta}/\partial u|_{(x_k,u_k)}$, and $c_k=g_{\theta}(x_k,u_k)-A_kx_k-B_ku_k$. The Jacobian-freezing mechanism uses $\hat{x}_{k+i+1}=A_k\hat{x}_{k+i}+B_k\hat{u}_{k+i}+c_k$ over a prediction horizon. For autonomous or fixed-input rollouts, local asymptotic stability is predicted when $\rho(A_k)<1$, where $\rho$ is the spectral radius; $\|A_k\|_2<1$ is a sufficient contraction condition. The first neglected nonlinear term gives local mismatch of order $O(\|\delta x\|^2+\|\delta u\|^2)$.

## Key formulas

- $$A_k=\left.\frac{\partial g_{\theta}(x,u)}{\partial x}\right|_{(x_k,u_k)},\qquad B_k=\left.\frac{\partial g_{\theta}(x,u)}{\partial u}\right|_{(x_k,u_k)},\qquad c_k=g_{\theta}(x_k,u_k)-A_kx_k-B_ku_k.$$
- $$\widehat{x}_{k+i+1}=A_k\widehat{x}_{k+i}+B_k\widehat{u}_{k+i}+c_k,\qquad i=0,\ldots,H-1.$$
- $$\rho(A_k)<1\ \Longrightarrow\ \|A_k^i\|\to0;\qquad \mathcal{L}_{\mathrm{jac}}=\left[\max\left(0,\rho(A_k)-r\right)\right]^2,$$
- $$g_{\theta}(x_k+\delta x,u_k+\delta u)=g_{\theta}(x_k,u_k)+A_k\delta x+B_k\delta u+O(\|\delta x\|^2+\|\delta u\|^2).$$

## Implementation notes

Integrate this at the rollout layer of a recurrent world model, neural state-space model, or SSM that predicts latent states. At the beginning of every training sequence, compute one Jacobian-vector-product-based estimate of $A_k$ and $B_k$ at the observed initial state; construct $c_k$; then use the affine recurrence for an auxiliary horizon loss or for inference-time planning. Pseudocode: (1) evaluate $y=g_\theta(x_k,u_k)$; (2) obtain $A_k$ and $B_k$ with automatic differentiation, Hutchinson/vector-Jacobian products, or an explicit Jacobian for small latent dimensions; (3) set $c_k=y-A_kx_k-B_ku_k$; (4) repeatedly update $\hat{x}\leftarrow A_k\hat{x}+B_ku+c_k$; (5) minimize prediction loss plus $\lambda[\max(0,\rho(A_k)-r)]^2$. Compute $\rho(A_k)$ exactly for small states, or estimate it with 10--20 power iterations; use $r<1$, such as $0.98$, as the target. The paper supplies the frozen-affine construction; the contraction penalty and neural rollout use are adaptations. The first cheap experiment is a GRU or MLP state-space model trained on damped nonlinear oscillator or Lorenz-63 sequences, comparing ordinary nonlinear rollout, frozen-affine rollout, and spectral-penalty training. Sweep the learned Jacobian gain and measure the divergence boundary. The quantitative prediction is that affine error grows quadratically with radius around the expansion state, while long-horizon error sharply worsens when measured $\rho(A_k)$ crosses $1$; the empirical boundary should be within roughly 20% of this predicted threshold.

## Disclaimer

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