[论文解读] Solving Inverse Physics Problems with Score Matching

Benjamin Holzschuh, Simona Vegetti+1 more|arXiv (Cornell University)|Jan 24, 2023

Gaussian Processes and Bayesian Inference被引用 7

一句话总结

论文提出了 SMDP，一种受扩散启发的方法，使用反向物理模拟器和学习修正来解决随时间演化的物理系统中的逆问题，与评分匹配与似然训练相关理论联系紧密。

ABSTRACT

We propose to solve inverse problems involving the temporal evolution of physics systems by leveraging recent advances from diffusion models. Our method moves the system's current state backward in time step by step by combining an approximate inverse physics simulator and a learned correction function. A central insight of our work is that training the learned correction with a single-step loss is equivalent to a score matching objective, while recursively predicting longer parts of the trajectory during training relates to maximum likelihood training of a corresponding probability flow. We highlight the advantages of our algorithm compared to standard denoising score matching and implicit score matching, as well as fully learned baselines for a wide range of inverse physics problems. The resulting inverse solver has excellent accuracy and temporal stability and, in contrast to other learned inverse solvers, allows for sampling the posterior of the solutions.

研究动机与目标

• Motivate solving inverse problems for time-evolving physical systems where only end states are known.
• Develop a method that combines an approximate inverse physics simulator with a learned correction to invert dynamics.
• Provide theoretical connections between the learned correction and score matching / probability flow concepts.
• Demonstrate robustness, accuracy, and posterior sampling capabilities across diverse inverse-physics tasks.

提出的方法

• Model the forward physics with a stochastic differential equation driven by a physics simulator P and diffusion g(t).
• Introduce a differentiable reverse physics step tilde{P}^{-1} and a neural correction s_theta(x,t) to predict previous states (Equation 2).
• Train s_theta using 1-step loss (Equation 3) or multi-step sliding-window loss (Equation 4) to improve trajectory corrections.
• Show that 1-step training approximates score matching as Delta t -> 0 (Theorem 3.1), and multi-step training maximizes a variational lower bound for likelihood via the probability-flow ODE (Theorem 3.2).
• Relate inference to a neural SDE (Equation 9) or an ODE (probability flow) variant for sampling the posterior or obtaining a maximum-likelihood trajectory.
• Provide practical guidance on training schedule (S from 2 up to S_max) and inference settings (SMDP SDE vs SMDP ODE).

实验结果

研究问题

• RQ1 Can we solve inverse, time-evolving physics problems by combining a reverse approximate physics step with a learned correction?
• RQ2 Does the learned correction s_theta effectuate score matching for the forward physics SDE and how does multi-step training relate to likelihood training?
• RQ3 How do 1-step vs multi-step training impact accuracy, stability, and posterior sampling for inverse physics problems?
• RQ4 What are the empirical benefits of incorporating a differentiable inverse physics step compared to fully learned baselines across diverse physics domains?

主要发现

Method	p_flow_ODE_100%	p_flow_ODE_10%	p_flow_ODE_1%	p_flow_SDE_100%	p_flow_SDE_10%	p_flow_SDE_1%
multi-step	0.97	0.91	0.81	0.99	0.94	0.85
1-step	0.78	0.44	0.41	0.93	0.71	0.75
ISM	0.19	0.15	0.01	0.92	0.94	0.52
SSM-VR	0.17	0.49	0.27	0.88	0.94	0.67

• The proposed SMDP framework achieves accurate and temporally stable reconstructions for inverse physics problems and enables posterior sampling.
• 1-step training aligns with score matching as Delta t → 0, justifying the approach theoretically.
• Multi-step training corresponds to maximum likelihood training of the probability-flow ODE, improving trajectory stability.
• Across experiments (1D toy SDE, stochastic heat equation, buoyancy-driven flow, turbulence with unknown dynamics), SMDP with multi-step loss outperforms denoising/implicit score matching baselines and fully learned surrogates.
• The ODE variant yields deterministic, maximum-likelihood trajectories, while the SDE variant provides diverse samples and better spectral fidelity due to stochasticity.