Penalty-Free Natural Deep Ritz Method Based on de Rham Complex for High-Dimensional Dirichlet Boundary Value Problems

Jul 16, 2026

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Deep neural networks offer strong potential for high-dimensional PDEs, but enforcing Dirichlet essential boundary conditions is difficult, and penalty-based formulations require problem-specific retuning as dimension grows. This work develops a unified Natural Deep Ritz Method (NatDRM) for all d≥2 using the de Rham complex and a penalty-free boundary decomposition. Dirichlet constraints become three coupled natural Neumann-type subproblems with Ritz-type losses, avoiding any boundary penalty parameter. Discrete dimension-unified losses and gauge-fixing regularization ensure well-posedness; numerical tests up to 6D match or exceed optimally tuned DRM and PINN and stabilize where penalized DRM fails.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages23

Reading Time58 min

Introduction

Background and motivation

Dirichlet boundary enforcement challenges

Penalty-based DRM and PINN

Natural Deep Ritz Method (NatDRM)

arXiv :2607 .00676v1 [math .NA] 1 Jul 2026 PENALTY-FREE NATURAL DEEP RITZ METHOD BASED ON DE RHAM COMPLEX FOR HIGH-DIMENSIONAL DIRICHLET BOUNDARY VALUE PROBLEMS∗ JIARONG CHEN†, XIA JI†‡, HAIJUN YU¶§ , AND SHUO ZHANG§¶ Abstract. Deep neural networks show great promise for high-dimensional PDEs, yet enforcing essential boundary conditions remains challenging, especially as penalty parameters require problemspecific retuning with increasing dimensionality. In this work, we extend the Natural Deep Ritz Method (NatDRM) [H. Yu and S. Zhang, J. Comput. Phys., 537 (2025)] to a unified framework for all dimensions d ≥ 2 based on the de Rham complex and its penalty-free boundary decomposition: curl-type operators act on scalar potentials in 2D, vector potentials in 3D, and antisymmetric second-order tensor potentials in d ≥ 4, respectively. This method converts Dirichlet constraints into three coupled natural (Neumann-type) subproblems with corresponding Ritz-type losses, eliminating the need for a boundary penalty parameter β . We derive dimension-unified discrete losses, lightweight boundary-based gauge-fixing regularizations to resolve curl-kernel non-uniqueness, and a joint training procedure; extensions to variable-coefficient elliptic and semilinear Poisson problems are formulated at the first subproblem level. Numerical experiments on smooth benchmarks up to 6D show that NatDRM, without any penalty tuning, matches or exceeds the accuracy of optimally tuned DRM and PINN in most cases. It converges stably in 6D where penalized DRM fails formost penalty values, and exhibits synchronous decay of interior and boundary errors, resolving the inherent imbalance of penalty-based methods. Key words. High dimensional PDE, Deep neural network, Essential boundary value problem, Deep Ritz method MSC codes. 65N30, 65N12, 68T07, 35J25, 58A15 1. Introduction. In recent years, the use of deep neural networks (DNNs) for solving partial differential equations (PDEs) has emerged as one of the most active research directions in computational science [11, 15, 9, 10, 18] . Traditional numerical methods, such as finite element and finite difference schemes, while robust and theoretically well-founded, often suffer from the curse of dimensionality, wherein computational cost grows exponentially with the dimension of the problem domain [8, 20] . In contrast, deep learning-based approaches leverage the compositional structure of DNNs to mitigate this curse, enabling the treatment of high-dimensional PDEs previously deemed intractable. Foundational works, including the deep Ritz method (DRM) [6], physics-informed neural networks (PINNs) [17], deep Galerkin method (DGM) [19], and weak adversarial networks (WANs) [23], have achieved remarkable success in approximating solutions to both linear and nonlinear PDEs in high- ∗ Submitted to the editors DATE. Funding: This work was partially supported by the National Natural Science Foundation of China (NSFC) under grant numbers 92370205, 12494543, 12271512, and 12371389, and by the Strategic Priority Research Program of the Chinese Academy of Sciences under grant numbers XDA0480504 and XDB0640000 . †School of Mathematics and Statistics, Beijing Institute of Technology, 100081, Beijing, China ([3120241504@bit.edu.cn](3120241504@bit.edu.cn)). ‡Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, 100081, Beijing, China ([jixia@bit.edu.cn](jixia@bit.edu.cn)). § State Key Laboratory of Mathematical Sciences (SKLMS) and State Key Laboratory of Scientific and Engineering Computing (LSEC), Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China ([hyu@lsec.cc.ac.cn](hyu@lsec.cc.ac.cn)). ¶ School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China ([szhang@lsec.cc.ac.cn](szhang@lsec.cc.ac.cn)). 2 J. CHEN, X. JI, H. YU, AND S. ZHAN

FAQs

What problem does this paper address for deep neural network PDE solvers?

The paper targets enforcing essential Dirichlet boundary conditions in high-dimensional PDEs, where boundary handling becomes increasingly hard and penalty strategies require sensitive tuning.

How does the proposed NatDRM remove the need for a penalty parameter?

Using the de Rham complex and a penalty-free boundary decomposition, Dirichlet constraints are converted into three coupled natural (Neumann-type) subproblems with Ritz-type losses, eliminating the boundary penalty parameter β.

How are the boundary decomposition types selected across dimensions?

The method uses curl-type scalar potentials for 2D, vector potentials for 3D, and antisymmetric second-order tensor potentials for d≥4, yielding a unified framework for all dimensions d≥2.

Jul 16, 2026

Upload by Patrick