# Score-Based Stabilization for Time-Dependent Problems

arXiv: https://arxiv.org/abs/2607.25119 · HTML: https://mlsanity.com/paper/2607.25119/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 27 Jul 2026.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Score-Based Point Cloud Denoising](https://mlsanity.com/paper/2107.10981/) (2021)
- [Estimation of ordinary differential equation models with discretization error quantification](https://mlsanity.com/paper/1907.10565/) (2019)
- [Quantifying uncertainty in the numerical integration of evolution equations based on Bayesian isotonic regression](https://mlsanity.com/paper/2411.08338/) (2024)
- [Joint Bayesian Inference of Parameter and Discretization Error Uncertainties in ODE Models](https://mlsanity.com/paper/2511.23010/) (2025)

## Abstract (arXiv)

We propose a score-based stabilization framework for numerical simulation of partial differential equations, in which a learned score model defines a stabilization operator applied to provisional numerical updates. This operator augments standard time-stepping schemes by enforcing structure and physical consistency through a correction that drives iterates toward the manifold of admissible states. We show that the stabilization operator acts as a contraction toward this manifold, yielding a correction mechanism with basin-conditional stability. Numerical experiments on Advection, Korteweg-de Vries (KdV), Nonlinear Schrodinger (NLS), and Burgers' equations demonstrate improved robustness, suppression of nonphysical instabilities, and preservation of qualitative dynamics.