【專題演講】2025-01-06 10:00-11:00 Finite difference-guided Deep Learning methods for solving Hamilton-Jacobi equations Yen-Hsi Richard Tsai (Department of Mathematics and Oden Institute for Computational Engineering and Sciences, UT-Austin)
數學跨領域研究中心 2025年專題演講
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DATE |
2025-01-06 10:00-11:00
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PLACE |
數學系館 3F會議室
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SPEAKER |
Yen-Hsi Richard Tsai
(Professor Department of Mathematics and Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin)
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| TITLE |
Finite difference-guided Deep Learning methods for solving Hamilton-Jacobi equations
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ABESTRACT |
We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi~(HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based algorithm to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility. |
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國立成功大學數學系、國立成功大學數學跨領域研究中心
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