Self-Consistent Velocity Matching of Probability Flows

• URL: http://arxiv.org/abs/2301.13737v4
• Date: Tue, 14 Nov 2023 04:44:08 GMT
• Title: Self-Consistent Velocity Matching of Probability Flows
• Authors: Lingxiao Li, Samuel Hurault, Justin Solomon
• Abstract summary: We present a discretization-free scalable framework for solving a class of partial differential equations (PDEs) The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent. We use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance.
• Score: 22.2542921090435
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the probability flow characterized by the same velocity field. Instead of directly minimizing the residual of the fixed-point equation with neural parameterization, we use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wider range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves superior performance in high dimensions with less training time compared to alternatives.

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