We present a highly efficient methodology for solving Hamilton–Jacobi (HJ) PDEs and demonstrate its effectiveness in the context of optimal transport (OT). Our approach begins with the derivation of a novel implicit solution formula for HJ PDEs, grounded in the method of characteristics and closely related to classical Hopf-type formulations. Building upon this foundation, we propose a deep learning-based framework that computes viscosity solutions without reliance on supervised training data. By leveraging the mesh-free nature and expressive capacity of neural networks, the proposed method enables scalable, accurate, and computationally efficient solutions to high-dimensional and nonconvex HJ PDEs.
Furthermore, we show that this framework enables the construction of efficient OT models. By exploiting the characteristic structure of HJ PDEs, we demonstrate that the bidirectional OT map admits a closed-form representation through the solution of an associated HJ equation. Leveraging this insight, we develop a deep learning model that directly computes OT maps using the implicit solution formula, thereby eliminating the need for numerical integration of ODEs. This characteristic-driven formulation leads to substantial improvements in both the accuracy of the computed transport maps and the efficiency of the sampling process.
Copyright. All Rights Reserved.
