Use of neural networks to solve partial differential equations (PDEs) was introduced by Lagaris et al. (1998), but only over the past few years have we seen a surge in the applications of deep neural networks (two or more hidden layers) for the solution of low- and high-dimension PDEs over bounded domains in R^d. This has been driven by two major contributions: (1) Raissi et al. (2019), who referred to the approach as Physics-Informed Neural Networks (PINNs), and used a collocation approach to solve forward and inverse problems, and (2) E and Yu (2018) (also see arXiv) who proposed a deep Ritz (variational) formulation to solve boundary-value problems. PINNs have approximation power that can recover h-, p- and r-adaptive finite element solutions and they are well-suited to solve forward, inverse, parametric design and high-dimensional problems, which makes them a powerful and attractive choice.

From my prior work on meshfree methods and well-known issues pertaining to the satisfaction of essential boundary conditions in meshfree Galerkin methods, it stands to reason that this is also pertinent in PINNs. With an eye on solving solid continua problems over complex geometries, we introduced an approach to exactly impose boundary conditions in PINNs that is based on approximate distance functions (ADFs) and the theory of R-functions. The PINN ansatz is formed so that all boundary conditions for scalar PDEs are met—this improves network training and accuracy of the PINN solution.

Publications

N. Sukumar and R. Roy (2026), "A Wachspress-based Transfinite Formulation for Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Computational Mechanics, in press.

N. Sukumar and A. Acharya (2025), "Variational Formulation Based on Duality to Solve Partial Differential Equations: Use of B-splines and Machine Learning Approximants," Computer Methods in Applied Mechanics and Engineering, Vol. 441, Article 117909.

R. J. Gladstone, M. A. Nabian, N. Sukumar, A. Srivastava and H. Meidani (2025), "FO-PINNs: A First-Order Formulation for Physics Informed Neural Networks," Engineering Analysis with Boundary Elements, Vol. 174, Article 106161.

S. Berrone, C. Canuto, M. Pintore and N. Sukumar (2023), "Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks and Variational Physics-Informed Neural Networks," Heliyon, Vol. 9, Article e18820.

N. Sukumar and A. Srivastava (2022), "Exact Imposition of Boundary Conditions with Distance Functions in Physics-Informed Deep Neural Networks," Computer Methods in Applied Mechanics and Engineering, Vol. 389, Article 114333. This method has been implemented in NVIDIA Modulus (April 2022 Release).

Presentations

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks (with R. Roy)," WCCM-ECCOMAS 2026, Munich, Germany, July 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Invited Seminar, Department of Engineering Science, Oxford University, Oxford, UK, July 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks (with R. Roy)," Congress on Numerical Methods in Engineering (CMN 2026), Gijón, Spain, July 2026.

• "Generalized Barycentric Coordinates in Computational Mechanics: From VEM to PINNs," Invited Lecture, MS on Generalized Barycentric Coordinates (Organizer: Kai Hormann), Curves and Surfaces 2026, Saint-Malo, France, June 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Distinguished Scholar Lecture (Keh-Chih Hwang Lecture Series), School of Aerospace Engineering, Tsinghua University, Beijing, China, April 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Invited Seminar, University of Chinese Academy of Sciences, Beijing, China, April 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Invited Seminar, School of Mechanics and Engineering Science, Peking University, Beijing, China, April 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Invited Seminar, Department of Engineering Mechanics, Dalian University of Technology, Dalian, China, April 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Invited Seminar, Department of Mathematics, University of California, Davis, CA, April 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks," Graduate Seminar, Department of Civil and Systems Engineering, Johns Hopkins University, Baltimore, MD, March 2026.

• "A Wachspress-Based Transfinite Formulation For Exactly Enforcing Dirichlet Boundary Conditions on Convex Polygonal Domains in Physics-Informed Neural Networks (with R. Roy)," Invited Seminar, CRUNCH Group Seminar, Brown University, Providence, RI, February 2026.

• "Variational Formulation Based on Duality to Solve Partial Differential equations: Use of B-splines and Machine Learning Approximants (with A. Acharya)," SIAM CSE 2025, Fort Worth, TX, March 2025.

• "Variational Formulation Based on Duality to Solve Partial Differential equations: Use of B-splines and Machine Learning Approximants," Invited Seminar, CRUNCH Group Seminar, Brown University, Providence, RI, February 2025.

• "Solving Partial Differential Equations with Physics-Informed Neural Networks Based on a Dual Variational Principle (with A. Acharya)," WCCM 2024/PANACM 2024, Vancouver, Canada, July 2024.

• "Exact Imposition of Boundary Conditions in PINNs to Solve PDEs," Invited Seminar, Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA, November 2023.

• "Exact Imposition of Boundary Conditions in PINNs to Solve PDEs," Invited Seminar, Department of Applied Mathematics, University of Waterloo, Ontario, Canada, October 2023.

• "Use of Generalized Barycentric Maps to Exactly Impose Dirichlet Boundary Conditions on Convex Geometries in Physics-Informed Deep Neural Networks," 2023 SES Annual Technical Meeting, Minneapolis, MN, October 2023.

• "Exact Imposition of Boundary Conditions in PINNs to Solve PDEs," Invited Lecture, BIRS Scientific Machine Learning Workshop, Banff, Canada, June 2023.

• "Recent Advances in Exact Imposition of Boundary Conditions in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Department of Mechanical Engineering, Boston University, Boston, MA, May 2023.

• "Recent Advances in Exact Imposition of Boundary Conditions in Physics-Informed Deep Neural Networks to Solve PDEs," Rhodes Information Initiative Seminar, Duke University, Durham, NC, May 2023.

• "Recent Advances in Exact Imposition of Boundary Conditions in Physics-Informed Deep Neural Networks to Solve PDEs," Joint Materials and Mechanics Seminar, Brown University, Providence, RI, May 2023.

• "Recent Advances in Exact Imposition of Boundary Conditions in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Dassault Systemes Simulia Corporation, Johnston, RI, March 2023.

• "Exact Imposition of Boundary Condition in Physics-Informed Deep Neural Networks to Solve PDEs," SEMM Seminar, Department of Civil & Environmental Engineering, University of California, Berkeley, CA, October 2022.

• "Exact Imposition of Boundary Condition in Physics-Informed Deep Neural Networks to Solve PDEs (with A. Srivastava)," USACM Thematic Conference on Meshfree and Novel Finite Element Methods, Berkeley, CA, September 2022.

• "Recent Advances in Exact Imposition of Boundary Condition in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Center for Machine Intelligence and Data Science, IIT Bombay, Mumbai, India, September 2022.

• "Recent Advances in Exact Imposition of Boundary Condition in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Department of Computational and Data Sciences, Indian Institute of Science, Bengaluru, India, July 2022.

• "Recent Advances in Polyhedral Virtual Element Methods and Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Department of Mechanical Engineering, Indian Institute of Science, Bengaluru, India, July 2022.

• "Recent Advances in Polyhedral Virtual Element Methods and Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, T-3 Division, Los Alamos National Laboratory, Los Alamos, NM, May 2022.

• "Recent Advances in Polyhedral Virtual Element Methods and Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Engineering Sciences Center, Sandia National Laboratories, Albuquerque, NM, May 2022.

• "Exact Imposition of Boundary Conditions with Distance Functions in Physics-Informed Deep Neural Networks," Invited Seminar, Mechanics and Computation Seminar, Stanford University, Stanford, CA, February 2022.

• "Meshfree Analysis on Complex Geometries Using Physics-Informed Deep Neural Networks," Invited Seminar, Instituto Superior Ténico, University of Lisbon, Portugal, January 2022.

• "Meshfree Analysis on Complex Geometries Using Physics-Informed Deep Neural Networks," Invited Seminar, Sandia National Laboratories, Albuquerque, NM, June 2021.

• "Exact Imposition of Boundary Conditions with Distance Functions in Physics-Informed Deep Neural Networks (with A. Srivastava)," CRUNCH Group Seminar, Brown University, Providence, RI, May 2021.

Useful Links

• Machine Learning + X (Crunch Group) Seminars at Brown University

• NVIDIA Modulus^™ Toolkit