Solving Inverse Physics Problems with Score Matching (2301.10250v2)

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.

• The paper's main contribution is its novel score matching approach that combines a learned correction function with an approximate inverse simulator to reverse stochastic dynamics.
• The methodology employs single-step score matching and multi-step maximum likelihood losses to enhance the stability and accuracy of reconstructing physical trajectories.
• Experimental results on quadratic SDEs and stochastic heat equations demonstrate the framework's robustness in recovering initial states under uncertainty.

Solving Inverse Physics Problems with Score Matching

The paper "Solving Inverse Physics Problems with Score Matching" introduces an innovative approach for addressing inverse problems in physical systems by leveraging advancements in diffusion models. It proposes moving the system's current state backward in time through a combination of an approximate inverse physics simulator and a learned correction function. This novel methodology integrates recent developments in score matching and maximum likelihood training to enhance accuracy and stability in reversing the simulation of physical dynamics.

Methodology and Contributions

The core contribution of the paper lies in its unique application of score matching techniques to inverse physics problems. The approach operates within a framework where the forward time evolution of a system is modeled as a stochastic differential equation (SDE). By designing a reverse-time SDE, the methodology aims to reconstruct initial states from observed end states, a task critical for problems requiring the reconstruction of the trajectory of a physical system from final observations.

The authors propose the combination of a learned correction function, denoted as $s_\theta$, and an approximate inverse of the physics simulator, $\Tilde{\mathcal{P}^{-1}$. These elements work together to adjust predictions of previous states, correcting for errors that arise due to the approximation of the inverse physics and noise in the data. This allows for the prediction of prior states under uncertainty, thereby enabling sampling from the posterior distribution of initial states.

The training framework leverages both single-step and multi-step losses to optimize $s_\theta$. The single-step loss is shown to minimize a score matching objective directly, while the multi-step loss corresponds to maximum likelihood training for a related neural ordinary differential equation. This multi-step approach enhances the stability and accuracy of the predicted trajectories by considering the interactions over several time steps, which is crucial for the integrity of the reverse simulation.

Experimental Evaluation

Several experiments highlight the efficacy of this method across diverse inverse physics problem domains. In the 1D toy example that employs a quadratic SDE, the paper demonstrates the improved posterior sampling achieved by the proposed technique compared to established score matching baselines like implicit score matching (ISM) and sliced score matching (SSM-VR). The results affirm that the combination of learned corrections and reverse time simulation can accurately recover initial states and match the known posterior distribution effectively.

Subsequent experiments utilize the stochastic heat equation and buoyancy-driven flow to test the approach's performance on more complex systems. These tests illustrate the robustness of the method in handling scenarios where information might be lost or heavily perturbed due to stochastic influences or non-linear dynamics, respectively. Remarkably, the neural network representation of $s_\theta$ in these contexts demonstrates its capability to learn intricate correlations and adjustments required for precise state reconstructions.

Implications and Future Work

The proposed method lays a strong theoretical and practical foundation for solving inverse problems in physics, wherein the goal is to retrieve historical data or system states based on present observations. This capability is crucial for fields like astrophysics, climate modeling, and any domain where system dynamics need to be inferred backward in time under uncertainty.

Future research could focus on several enhancements, including reducing computational costs associated with extensive backward simulation and exploring reduced-order modeling techniques. Additionally, further work could investigate extending the methodology to broader classes of PDEs and exploring synergies with other machine learning paradigms for improved generalization capabilities. Such advancements could elevate the utility of this framework in scientific and engineering applications where predictive accuracy and stability in deducing historical states are paramount.