Deep Tangent Bundle (DTB) method: a Deep Neural Network approach to compute solutions of PDES

2025/08/31 by Hao Wu, Haomin Zhou, Wu, Hao +1
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Neural Networks and Reservoir Computing #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2509.00957

openalex publication_date 2025/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop a numerical framework, the Deep Tangent Bundle (DTB) method, that is suitable for computing solutions of evolutionary partial differential equations (PDEs) in high dimensions. The main idea is to use the tangent bundle of an adaptively updated deep neural network (DNN) to approximate the vector field in the spatial variables while applying the traditional schemes for time discretization. The DTB method takes advantage of the expression power of DNNs and the simplicity of the tangent bundle approximation. It does not involve nonconvex optimization. Several numerical examples demonstrate that the DTB is simple, flexible, and efficient for various PDEs of higher dimensions.

Citations

• A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations
• Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method
• Structured and Balanced Multi-Component and Multi-Layer Neural Networks
• Parameterized Wasserstein Gradient Flow
• Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks
• Parameterized Wasserstein Hamiltonian Flow
• Neural Control of Parametric Solutions for High-dimensional Evolution PDEs
• Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations
• Fourier Neural Operator for Parametric Partial Differential Equations
• Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
• Wasserstein Hamiltonian flows
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems

Related

• Positional Embeddings for Solving PDEs with Evolutional Deep Neural Networks
• Partial Differential Equations Meet Deep Neural Networks: A Survey
• Learning Neural PDE Solvers with Convergence Guarantees
• Low-Rank Evolutionary Deep Neural Networks via Adaptive Tangent-Space Reduction
• Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations