機器學習在偏微分方程式的應用

This course provides an introduction to deep neural networks and physics-informed neural networks (PINNs) from the perspective of approximation theory and scientific computing. The course begins with neural networks as tools for function approximation, emphasizing their mathematical foundations, approximation properties, and the role of depth, width, and activation functions. Fundamental concepts in deep learning, including optimization, backpropagation, automatic differentiation, and generalization, will then be introduced. Building on these foundations, the course explores physics-informed neural networks (PINN), in which physical laws described by ordinary and partial differential equations are incorporated into the training of neural networks. Students will learn how neural networks can be used to approximate solutions of differential equations and how differential operators, initial and boundary conditions, observational data, and physical constraints can be incorporated into loss functions. The course will also examine the mathematical and computational challenges of PINNs, including approximation and optimization errors, training difficulties, spectral bias, loss balancing, collocation-point sampling, and the accurate approximation of derivatives. Connections with classical numerical methods for differential equations will be emphasized throughout the course. Recent developments in scientific machine learning will also be introduced, including adaptive PINNs, domain-decomposition methods, neural operators, physics-informed operator learning, advanced neural architectures, inverse problems, parameter identification, and hybrid approaches combining machine learning with traditional numerical methods.

Linear Algebra and Analysis

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blackboard

The grade will be based on the following: 1. Final oral project: 40% 2. Two oral exams: 40%+ 20%

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lectures

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聯絡方式

make appointments and email