Solving physics-constrained inverse problems with conditional flow matching

Abstract

This study presents a conditional flow matching framework for solving physics-constrained Bayesian inverse problems. In this setting, samples from the joint distribution of inferred variables and measurements are assumed available, while explicit evaluation of the prior and likelihood densities is not required. We derive a simple and self-contained formulation of both the unconditional and conditional flow matching algorithms, tailored specifically to inverse problems. In the conditional setting, a neural network is trained to learn the velocity field of a probability flow ordinary differential equation that transports samples from a chosen source distribution directly to the posterior distribution conditioned on observed measurements. This black-box formulation accommodates nonlinear, high-dimensional, and potentially non-differentiable forward models without restrictive assumptions on the noise model. We further analyze the behavior of the learned velocity field in the regime of finite training data. Under mild architectural assumptions, we show that overtraining can induce degenerate behavior in the generated conditional distributions, including variance collapse and a phenomenon termed selective memorization, wherein generated samples concentrate around training data points associated with similar observations. A simplified theoretical analysis explains this behavior, and numerical experiments confirm it in practice. We demonstrate that standard early-stopping criteria based on monitoring test loss effectively mitigate such degeneracy. The proposed method is evaluated on several physics-based inverse problems. We investigate the impact of different choices of source distributions, including Gaussian and data-informed priors. Across these examples, conditional flow matching accurately captures complex, multimodal posterior distributions while maintaining computational efficiency.

Figures (22)

• Figure 1: Visualizing Case 1 corresponding to \ref{['eq:overfit-case-1']}
• Figure 2: Visualizing Case 2 corresponding to \ref{['eq:overfit-case-2']}
• Figure 3: (a) Train and test data for the toy example used to illustrate effects of overfitting. (b) Train and test loss curve for the velocity network trained using the training data shown in (a). The blue curve above shows the moving average (MA) of the test loss. The MA is computed over a window of size 500
• Figure 4: Kernel density estimates of the conditional distribution $\rho_{X|Y}(x \mid y = 0.6)$ estimated from samples generated using the trained velocity network at different stages of training
• Figure 5: Mean and one-standard-deviation interval of the conditional distribution $\rho_{X|Y}$ estimated using samples generated by the trained velocity network at different stages of training

Theorems & Definitions (4)

• Proposition 2.1
• proof
• Proposition 2.2
• proof