Score-Based Stabilization for Time-Dependent Problems

Data-driven surrogate models can significantly accelerate the simulation of continuous dynamical systems, yet the step-wise accumulation of errors during autoregressive time-stepping often leads to spectral blow-up and unphysical divergence. Existing global regularization techniques can enforce contractive dynamics but uniformly damp high-frequency features, causing over-smoothing; meanwhile, long-horizon trajectory optimization methods are severely constrained by memory bottlenecks. This paper proposes Jacobian-Adaptive Weighting for Stability (JAWS), which reformulates operator learning as a Maximum A Posteriori (MAP) estimation problem with spatially heteroscedastic uncertainty, enabling the regularization strength to adapt automatically based on local physical complexity: enforcing contraction in smooth regions to suppress noise while relaxing constraints near singular features such as shocks to preserve gradient information. Experiments demonstrate that JAWS serves as an effective spectral pre-conditioner for trajectory optimization, allowing short-horizon, memory-efficient training to match the accuracy of long-horizon baselines. Validations on the 1D viscous Burgers' equation and 2D flow past a cylinder ($\text{Re}=400$ out-of-distribution generalization) confirm the method's advantages in long-term stability, preservation of physical conservation properties, and computational efficiency. This significant reduction in memory usage makes the method particularly well-suited for stable and efficient long-term simulation of large-scale flow fields in practical engineering applications.

We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains $\mathcal{O}(τ)$, yielding a Gronwall stability factor bounded uniformly in the step size and independent of the spatial resolution. The network is trained end-to-end through a solver-in-the-loop objective that unrolls the full iteration and penalises trajectory error in a Bourgain-type norm, aligning learning with multi-step solver dynamics rather than isolated one-step targets. Under stated assumptions, the global error satisfies $C(\varepsilon_{net}+δ)\,τ^γ\ln(1/τ)$, where $\varepsilon_{net}$ measures the network approximation quality and $δ$ the training shortfall. Experiments on three dispersive benchmarks with rough data show that HIN-LRI improves accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.

Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property, which we call Symmetry Rupture, serves two purposes. As a training regularizer, it confines the hypothesis space to flows that compose consistently across temporal scales. As an inference oracle, it lets the solver select the largest step size that preserves internal consistency, replacing the local truncation error that conventional adaptive solvers depend on. On the diffusion-reaction benchmark under time-informed inference, our method reduces rollout RMSE by 87% while using 5x fewer function evaluations than a Neural ODE baseline. In the more demanding direct auto-regressive setting, where the model must predict distant future frames without intermediate temporal cues, our adaptive solver allocates compute based on local geometric complexity -- maintaining the lowest rollout RMSE on two of three PDE benchmarks while baselines either diverge or require up to an order of magnitude more function evaluations to remain stable.