[論文レビュー] A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

Suchuan Dong, Yuchuan Zhang|arXiv (Cornell University)|Mar 23, 2026

Machine Learning and ELM被引用数 0

ひとこと要約

The paper presents a systematic method to exactly enforce Dirichlet, Neumann, and Robin boundary conditions on general quadrilateral domains with curved boundaries using exact domain mappings, Theory of Functional Connections (TFC), and transfinite interpolation, implemented with Extreme Learning Machine (ELM).

ABSTRACT

We present a systematic method for exactly enforcing Dirichlet, Neumann, and Robin type conditions on general quadrilateral domains with arbitrary curved boundaries. Our method is built upon exact mappings between general quadrilateral domains and the standard domain, and employs a combination of TFC (theory of functional connections) constrained expressions and transfinite interpolations. When Neumann or Robin boundaries are present, especially when two Neumann (or Robin) boundaries meet at a vertex, it is critical to enforce exactly the induced compatibility constraints at the intersection, in order to enforce exactly the imposed conditions on the joining boundaries. We analyze in detail and present constructions for handling the imposed boundary conditions and the induced compatibility constraints for two types of situations: (i) when Neumann (or Robin) boundary only intersects with Dirichlet boundaries, and (ii) when two Neumann (or Robin) boundaries intersect with each other. We describe a four-step procedure to systematically formulate the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on general quadrilateral domains. The method developed herein has been implemented together with the extreme learning machine (ELM) technique we have developed recently for scientific machine learning. Ample numerical experiments are presented with several linear/nonlinear stationary/dynamic problems on a variety of two-dimensional domains with complex boundary geometries. Simulation results demonstrate that the proposed method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries exactly, with the numerical boundary-condition errors at the machine accuracy.

研究の動機と目的

• Develop a systematic way to map general quadrilateral domains with curved boundaries to a standard domain for exact BC enforcement.
• Formulate trial functions that exactly satisfy Dirichlet, Neumann, and Robin conditions on curved boundaries.
• Handle induced compatibility constraints at intersections of Neumann/Robin boundaries, including vertex cases where two boundaries meet.
• Provide a four-step procedure to construct general forms of boundary-satisfying functions on general quadrilaterals.
• Demonstrate the method's effectiveness through numerical experiments on linear and nonlinear PDEs on complex geometries using ELM.

提案手法

• Use exact mappings between general quadrilateral domains and the standard domain via a combination of TFC constrained expressions and transfinite interpolation.
• Represent the mapping and trial functions in parametric form on the standard domain and enforce boundaries exactly through a constrained expression.
• Solve the resulting systems with Extreme Learning Machine (ELM) where hidden-layer coefficients are fixed and only output-layer coefficients are trained.
• Develop a four-step procedure to formulate trial functions that satisfy Dirichlet, Neumann, and Robin conditions and their vertex compatibility constraints.
• Address two Neumann/Robin interaction scenarios: (i) Neumann intersects only with Dirichlet, and (ii) two Neumann/Robin boundaries intersecting at a vertex.]
• research_questions: [

実験結果

リサーチクエスチョン

• RQ1 一般のい形四辺形領域を曲線境界を保持したまま標準領域へ写像して境界条件を正確に適用する方法はどう構築できるか？
• RQ2 Dirichlet、Neumann、およびRobin条件を曲線境界上で正確に課すには、境界の交差点での適合性を含めてどう扱うべきか？
• RQ3 一般の四辺形領域に imposed 境界条件を満たす試行関数を系統的に構築する四段階の手順は何か？
• RQ4 ELMを用いたとき、複雑な幾何形状上の線形・非線形PDEに対して提案手法はどの程度の精度を示すか？

主な発見

• 提案手法は曲線境界を持つ領域境界上でDirichlet、Neumann、およびRobin境界条件を厳密に適用し、数値実験で境界条件を機械精度で満たすことを示す。
• NeumannまたはRobin境界が交わる頂点での完全な適合性制約を計算・適用し、隣接境界条件の正確な満足を保証。
• imposed 境界条件とその適合性制約を満たす一般的な試行関数を構築するための四段階の手順を確立。
• Extrem Learning Machineで実装し、複雑な2次元領域の線形・非線形定常・動的PDEを多様な境界で検証。
• 結果は高い精度と安定性を示し、ELM以外の適用にも広く適用可能であることを示唆。