Robust Physics-Sparse Neural Dynamics

1) Integration point: implement this as the transition model in a small neural state-space model. Encode observations with an MLP encoder, maintain latent state \(x_k\), and predict \(x_{k+1}\) using the sparse hybrid equation rather than a fully unconstrained MLP. For a generic benchmark, use a known linear relaxation \(f_{\mathrm{known}}(x,u)=-\Gamma(x-x^*(u))\), analogous to the droop filter, where \(\Gamma\) is a positive diagonal matrix; use a library containing \(x_j\), \(u_l\), \(x_jx_m\), \(x_ju_l\), and bounded nonlinearities. 2) Pseudocode: estimate derivatives with a centered finite difference or automatic differentiation through a short encoder; form \(A=[\Theta\ \ -\dot X]\); repeatedly sample minimal subsets, solve the TLS problem by SVD, score inliers by normalized residual, retain the best consensus set, and apply iterative hard thresholding to remove coefficients below threshold \(\tau\). During neural training, alternate 5--20 sparse-regression updates with gradient updates of the encoder and add reconstruction loss, multi-step rollout loss, and \(\alpha\mathcal L_{\mathrm{stab}}\). Estimate \(\rho(J_k)\) with 10--20 power iterations rather than explicitly forming eigenvalues. 3) Computed from the paper's mechanism: the known relaxation structure, sparse library identification, TLS treatment of errors in both regressors and derivatives, and RANSAC outlier rejection. Estimated empirically: derivative noise scale, RANSAC inlier threshold, sparsity threshold \(\tau\), and the latent Jacobian spectrum. 4) First cheap experiment: train on noisy trajectories from a two-dimensional damped oscillator with 5% Gaussian observation noise and 5% randomly corrupted time points, comparing an unconstrained GRU/MLP transition, ordinary-least-squares SINDy, and the TLS-RANSAC hybrid model. Sweep integration step \(h\) and multiply the learned transition Jacobian by a scalar gain. The quantitative prediction is a sharp rollout-divergence boundary near \(\rho(J)=1\); the constrained model should remain bounded for gains with measured \(\rho(J)<1\), while divergence should occur predominantly after crossing one. On held-out trajectories with corrupted observations, TLS-RANSAC should have lower coefficient error and substantially longer stable rollout horizon than ordinary least squares. Report the measured critical gain and test whether it predicts the 100-step rollout boundary within 20%.