The study investigates how Jacobian-based fine‑tuning affects neural emulators of chaotic partial differential equations (PDEs). By decomposing error growth into physical and model‑induced components, the authors introduce an amplification ratio that quantifies how the emulator’s dynamics magnify prediction errors. Across 21 experimental setups, Jacobian fine‑tuning significantly reduced errors—especially for configurations with high amplification ratios—while matching the solver’s Jacobian along the model’s error direction (EDJM) outperformed random directions and input‑noise injection.
arXiv:2608. 16084v1 Announce Type: new Abstract: Neural autoregressive models have rapidly emerged as powerful emulators of high-dimensional chaotic systems, yet their long-term instability and error growth remain poorly understood, leading to ad-hoc solutions.
The paper introduces a method called model‑constrained randomized Jacobian matching to enforce second‑order consistency when learning chaotic dynamical systems. By comparing Jacobians at randomly perturbed inputs, the approach implicitly penalises Hessian mismatch without computing full Hessian tensors, achieving $O(d^2)$ memory cost. Experiments on Lorenz 63 and Lorenz 96 show that this second‑order supervision reduces invariant‑measure error, improves Lyapunov‑spectrum accuracy, and avoids spurious attractors that plague first‑order methods.
Neural surrogates for forecasting physical fields from sparse, noisy sensors typically use a spatial window and a temporal history of past snapshots. This study compares the trade‑off between window size and history length against the Bayes‑optimal error, showing that for fields with local dynamics the optimal forecast relies mainly on distant sensors via spatial correlation. Experiments on Burgers flow and the chaotic Kuramoto‑Sivashinsky equation reveal that the benefit of additional snapshots depends on lead time and system stability, leading to design rules that size the window in correlation lengths and split input budgets between window and history based on forecast horizon.
arXiv:2606. 01596v1 Announce Type: cross Abstract: Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics.
arXiv:2609.13204v1 Announce Type: new Abstract: Neural ordinary differential equations use numerical solvers inside the training loop. The solver determines the forward trajectory and also affects th...
arXiv:2507. 03631v4 Announce Type: replace Abstract: Extracting interpretable mathematical models from complex dynamical systems is difficult, especially for chaotic dynamics observed with noisy experimental data.
arXiv:2610.06798v2 Announce Type: replace-cross Abstract: Machine-learning emulators of chaotic and stochastic systems are usually validated on forecast skill and long-run statistics. Neither certifi...
Machine-learning emulators of chaotic and stochastic systems are usually validated on forecast skill and long-run statistics. Neither certifies that an emulator responds correctly to forcing, the property that projection and attribution studies rely on.
arXiv:2606. 09929v1 Announce Type: cross Abstract: Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable.
The paper introduces the Latent Generative Solver (LGS), a neural PDE solver that combines a Physics VAE, a Pyramidal Flow-Forcing Transformer, and input noising to achieve generalization across twelve PDE families and stable long-term rollouts. LGS matches or surpasses deterministic baselines on one-step predictions, outperforms them on 5- and 10-step rollouts, and significantly reduces long-horizon error while cutting compute costs. It also adapts efficiently to unseen higher-resolution systems, demonstrating strong empirical performance on 2D regular-grid PDE simulations.
Physics‑Informed Neural Networks (PINNs) often fail on stiff or advection‑dominated partial differential equations. This study compares two proposed fixes—switching from FP32 to FP64 precision to address an L‑BFGS stopping artifact, and replacing the MLP with a state‑space‑model (SSM) backbone plus sub‑sequence alignment to mitigate architectural simplicity bias—using a pre‑registered, seed‑paired 144‑run experiment across convection, reaction, and wave problems, plus an independent 85‑run study. The results show that the remedies act on disjoint subsets of regimes and seeds: precision changes affect some seeds in opposite directions, alignment improves success in hard convection cases, and the SSM backbone alone succeeds on many reaction seeds, but none of the remedies fully substitutes for the other, and all must be evaluated jointly and reported per seed.
arXiv:2606. 21828v2 Announce Type: replace-cross Abstract: Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction.