Rate of Approximation by Flows: A Case Study on the Eikonal Equation
Keywords: flow map, approximation rate, eikonal equation
Abstract: Previous works have demonstrated the universal approximation capability of residual networks through their continuous idealization as flow maps of dynamical systems. However, informative results on their approximation rates in terms of depth (corresponding to time) are generally lacking. From the viewpoint of approximation theory, a major difficulty in addressing this gap lies in identifying an appropriate target space for the approximation problem. In this paper, we introduce a restrictive but useful target function space comprised of solutions to the eikonal equations, a type of first-order nonlinear partial differential equation, to investigate the approximation rates of flow map families. We provide an estimate of the approximation error within this space, which is notably different from classical rate estimates based directly on the smoothness of target functions. This theoretical result further inspires a new learning-based algorithm for solving the eikonal equation. Experimental results validate the effectiveness of our proposed algorithm, including its robustness to spatial resolution and solution regularity, as well as transferability among similar problems.
Primary Area: learning theory
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Submission Number: 9934
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