A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

Suchuan Dong; Yuchuan Zhang

math.NAMar 23, 2026

A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

Authors: Suchuan Dong, Yuchuan Zhang

Organizations: Center for Computational and Applied Mathematics Department of Mathematics Purdue University, USA

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.

Figures & tables

Panel of Figure 1 — Figure 1: Mapping between a general quadrilateral domain $\Omega$ and the standard quadrilateral domain $\Omega_{st}=[-1,1]^{2}$ .

$\xi$	$\varphi_{0}(\xi)$	$\varphi_{0}^{\prime}(\xi)$	$\varphi_{1}(\xi)$	$\varphi_{1}^{\prime}(\xi)$	$\psi_{0}(\xi)$	$\psi_{0}^{\prime}(\xi)$	$\psi_{1}(\xi)$	$\psi_{1}^{\prime}(\xi)$
$-1$	1	0	0	0	0	1	0	0
1	0	0	1	0	0	0	0	1

Table 1: Interpolation properties of

C^{1}

Hermite interpolation polynomials defined on

\xi\in[-1,1]

$\xi$	$\rho_{0}(\xi)$	$\rho_{0}^{\prime}(\xi)$	$\rho_{0}^{\prime\prime}(\xi)$	$\rho_{1}(\xi)$	$\rho_{1}^{\prime}(\xi)$	$\rho_{1}^{\prime\prime}(\xi)$	$\upsilon_{0}(\xi)$	$\upsilon_{0}^{\prime}(\xi)$	$\upsilon_{0}^{\prime\prime}(\xi)$
$-1$	1	0	0	0	0	0	0	1	0
$1$	0	0	0	1	0	0	0	0	0
$\xi$	$\upsilon_{1}(\xi)$	$\upsilon_{1}^{\prime}(\xi)$	$\upsilon_{1}^{\prime\prime}(\xi)$	$\omega_{0}(\xi)$	$\omega_{0}^{\prime}(\xi)$	$\omega_{0}^{\prime\prime}(\xi)$	$\omega_{1}(\xi)$	$\omega_{1}^{\prime}(\xi)$	$\omega_{1}^{\prime\prime}(\xi)$
$-1$	0	0	0	0	0	1	0	0	0
$1$	0	1	0	0	0	0	0	0	1

Table 2: Interpolation properties of

C^{2}

Hermite interpolation polynomials defined on

\xi\in[-1,1]

Panel of Figure 2 — Figure 2: Helmholtz equation: Domain geometries (top row) and the exact solutions (bottom row), (a,f) domain #1, (b,g) domain #2, (c,h) domain #3, (d,i) domain #4, and (e,j) domain #5.

Panel of Figure 3 — Figure 3: Helmholtz equation (Dirichlet BCs on all boundaries): Distributions of the NN solutions (top row) and their point-wise absolute errors (bottom row) on the five domains. Simulation parameters: Domain #1, $R_{m}=4.62$ , $Q=70$ , $M=800$ ; Domain #2, $R_{m}=4.0$ , $Q=65$ , $M=800$ ; Domain #3, $R_{m}=4.57$ , $Q=65$ , $M=800$ ; Domain #4, $R_{m}=3.53$ , $Q=60$ , $M=800$ ; Domain #5, $R_{m}=4.17$ , $Q=65$ , $M=800$ .

	domain #1	domain #2	domain #3	domain #4	domain #5
max-error (domain)	$1.167E-5$	$1.115E-7$	$1.485E-6$	$1.026E-8$	$4.350E-9$
rms-error (domain)	$2.752E-6$	$3.071E-8$	$1.034E-7$	$2.727E-9$	$8.668E-10$
max DBC-error ( $\overline{AB}$ )	$8.882E-16$	$4.441E-16$	$8.882E-16$	$8.882E-16$	$4.441E-16$
rms DBC-error ( $\overline{AB}$ )	$1.051E-16$	$5.747E-17$	$1.034E-16$	$2.271E-16$	$1.696E-16$
max DBC-error ( $\overline{BC}$ )	$2.220E-16$	$0.0$	$8.882E-16$	$4.441E-16$	$8.882E-16$
rms DBC-error ( $\overline{BC}$ )	$5.747E-17$	$0.0$	$1.371E-16$	$8.999E-17$	$1.134E-16$

Table 3: Helmholtz equation (Dirichlet BC on all boundaries): maximum and rms NN-solution errors over the domain (

e_{max}^{\Omega}

e_{rms}^{\Omega}

), and the maximum and rms DBC errors on the four boundaries (

\varepsilon_{max}^{\overline{AB}}

\varepsilon_{rms}^{\overline{AB}}

\varepsilon_{max}^{\overline{BC}}

\varepsilon_{rms}^{\overline{BC}}

\varepsilon_{max}^{\overline{CD}}

\varepsilon_{rms}^{\overline{CD}}

\varepsilon_{max}^{\overline{AD}}

\varepsilon_{rms}^{\overline{AD}}

). Simulation parameters follow those of Figure 3 .

Panel of Figure 4 — Figure 4: Helmholtz equation (Neumann or Robin BC on boundaries): Distributions of the NN solutions (top row) and their point-wise absolute errors (bottom row) on domain #1. (a,d) Case #1: NBC on $\overline{BC}$ , and DBCs on the other boundaries. (b,e) Case #2: RBC ( $\alpha=1.0$ ) on $\overline{BC}$ , and DBCs on the other boundaries. (c,f) Case #3: NBCs on $\overline{BC}$ and $\overline{CD}$ , and DBCs on the other boundaries. Simulation parameters: $R_{m}=5.0$ , $Q=70$ , $M=950$ for all cases.

	Case #1	Case #2	Case #3
max solution-error (domain)	$4.536E-5$	$4.217E-5$	$8.295E-5$
rms solution-error (domain)	$3.775E-6$	$3.420E-16$	$8.818E-6$
max DBC-error ( $\overline{AB}$ )	$0.0$	$0.0$	$0.0$
rms DBC-error ( $\overline{AB}$ )	$0.0$	$0.0$	$0.0$
max NBC- or RBC-error ( $\overline{BC}$ )	$1.421E-14$	$1.421E-14$	$1.421E-14$
rms NBC- or RBC-error ( $\overline{BC}$ )	$3.614E-15$	$4.180E-15$	$4.154E-15$

Table 4: Helmholtz equation on domain #1 (Neumann or Robin BCs): maximum and rms NN-solution errors over the domain, and the maximum and rms boundary-condition (DBC, NBC, RBC) errors on the boundaries. Different cases correspond to those in Figure 4 .

Panel of Figure 5 — Figure 5: Nonlinear Helmholtz equation: Domain geometries (top row) and the exact solutions (bottom row), (a,e) domain #1, (b,f) domain #2, (c,g) domain #3, and (d,h) domain #4.

Panel of Figure 6 — Figure 6: Nonlinear Helmholtz equation (Dirichlet BC on all boundaries): Distributions of the NN solutions (top row) and their point-wise absolute errors (bottom row) on the four domains. Simulation parameters: (a,e) $R_{m}=4.0$ , $Q=50$ , $M=1000$ . (b,f) $R_{m}=4.5$ , $Q=55$ , $M=1000$ . (c,g,d,h) $R_{m}=5.0$ , $Q=60$ , $M=1000$ . $Q_{db}=8$ (see Remark 2.10 ) for all domains.

	domain #1	domain #2	domain #3	domain #4
max-error (domain)	$2.544E-9$	$8.948E-7$	$4.475E-7$	$4.808E-7$
rms-error (domain)	$4.359E-10$	$2.099E-7$	$7.637E-8$	$1.118E-7$
max DBC-error ( $\overline{AB}$ )	$0.0$	$0.0$	$0.0$	$0.0$
rms DBC-error ( $\overline{AB}$ )	$0.0$	$0.0$	$0.0$	$0.0$
max DBC-error ( $\overline{BC}$ )	$8.882E-16$	$4.441E-16$	$5.551E-16$	$4.441E-16$
rms DBC-error ( $\overline{BC}$ )	$2.303E-16$	$1.662E-16$	$2.075E-16$	$1.615E-16$

Table 5: Nonlinear Helmholtz equation (Dirichlet BC on all boundaries): maximum and rms NN-solution errors over the domain, and the maximum and rms DBC errors on the four boundaries. Simulation parameters follow those of Figure 6 .

Panel of Figure 7 — Figure 7: Nonlinear Helmholtz equation (Neumann or Robin BCs): Distributions of the NN solutions (top row) and their point-wise errors (bottom row) on domain #1. (a,d) Case #1: Neumann condition on $\overline{BC}$ and Dirichlet condition on the other boundaries. (b,e) Case #2: Robin condition on $\overline{BC}$ and Dirichlet condition on the other boundaries. (c,f) Case #3: Neumann condition on $\overline{BC}$ and $\overline{CD}$ , and Dirichlet condition on the other boundaries. Simulation parameters: (a,d) $R_{m}=4.0$ , $Q=60$ , $M=1000$ . (b,e) $R_{m}=3.5$ , $Q=60$ , $M=1000$ . (c,f) $R_{m}=2.5$ , $Q=60$ , $M=1000$ .

	Case #1	Case #2	Case #3
max solution-error (domain)	$2.133E-8$	$1.045E-8$	$4.709E-8$
rms solution-error (domain)	$2.498E-9$	$1.367E-9$	$7.885E-9$
max DBC-error ( $\overline{AB}$ )	$0.0$	$1.388E-16$	$0.0$
rms DBC-error ( $\overline{AB}$ )	$0.0$	$6.418E-17$	$0.0$
max NBC- or RBC-error ( $\overline{BC}$ )	$5.329E-15$	$6.217E-15$	$7.105E-15$
rms NBC- or RBC-error ( $\overline{BC}$ )	$1.540E-15$	$2.198E-15$	$1.994E-15$

Table 6: Nonlinear Helmholtz equation on domain #1 (Neumann or Robin BCs): maximum and rms NN-solution errors over the domain, and the maximum and rms boundary-condition (DBC, NBC, RBC) errors on different boundaries. The three cases correspond to those in Figure 7 for different types of boundary conditions.

Panel of Figure 8 — Figure 8: Heat equation on deforming/moving domains: space-time domain geometries (top row) and the exact solutions (bottom row). (a,d) Domain #1 (deforming spatial domain); (b,e) Domain #2 (moving spatial domain); (c,f) Domain # ( another moving spatial domain).

Panel of Figure 9 — Figure 9: Heat equation on deforming/moving domains: Distributions of the NN solutions (top row), and their point-wise absolute errors (bottom row) for domains #1 (a,d), #2 (b,e) and #3 (c,f). Simulation parameters: Domains #1 and #3, $R_{m}=4.0$ , $Q=100$ , $Q_{db}=7$ , $M=1000$ ; Domains #2, $R_{m}=5.5$ , $Q=100$ , $Q_{db}=7$ , $M=1000$ .

	domain #1	domain #2	domain #3
max solution-error (domain)	$6.083E-8$	$7.503E-5$	$1.736E-9$
rms solution-error (domain)	$1.712E-8$	$1.401E-6$	$3.208E-10$
max BC-error ( $\overline{AB}$ )	$0.0$	$0.0$	$0.0$
rms BC-error ( $\overline{AB}$ )	$0.0$	$0.0$	$0.0$
max BC-error ( $\overline{BC}$ )	$8.882E-16$	$8.882E-16$	$8.882E-16$
rms BC-error ( $\overline{BC}$ )	$1.983E-16$	$3.436E-16$	$3.743E-16$

Table 7: Heat equation on deforming/moving domains: the maximum and rms NN solution errors over the space-time domain and on the boundary

\overline{CD}

, and the maximum and rms boundary/initial condition errors on

\overline{AB}

\overline{BC}

, and

\overline{AD}

. Note that Dirichlet conditions are imposed on

\overline{AB}

\overline{BC}

and

\overline{AD}

of the space-time domain, and that no boundary condition is imposed on

\overline{CD}

. Simulation parameters follow those of Figure 9 .

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A Method for Exact Enforcement of Boundary Conditions on General Curved Triangular Domains: Algorithm and Application to Scientific Machine Learning

Suchuan Dong, Yuchuan Zhang

Center for Computational and Applied Mathematics Department of Mathematics Purdue University, USA

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We present a method for enforcing exactly Dirichlet, Neumann, and Robin conditions on general triangular domains with arbitrarily curved boundaries. The algorithmic constructions developed here complements those in a previous work for general quadrilateral domains. Our method combines (i) the exact mapping between a general curved triangular domain and the standard triangle, (ii) TFC constrained expressions, and (iii) Barnhill transfinite interpolations, to provide explicit closed-form trial ansatze that mathematically satisfy the imposed boundary conditions exactly. Corner compatibility induced by the boundary conditions is pivotal, and enforcing exactly the imposed boundary conditions hinges on the exact enforcement of the induced compatibility constraints. The corner compatibility associated with the acute vertices of the standard triangle and the coupling between gradient components when the hypotenuse corresponds to a Neumann/Robin boundary pose particular challenges. Two strategies are critical to resolving these issues, by reformulating conditional corner constraints into equivalent unconditional forms and by introducing a separate TFC-style auxiliary edge construction together with an associated linear solve on the hypotenuse. We analyze the corner compatibility constraints and present formulations for the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on curved triangular domains. The proposed method has been implemented using extreme learning machine type randomized neural networks for solving PDE problems. We present ample numerical experiments with linear/nonlinear PDE problems on a variety of domains with complex boundary geometries. Numerical results show that the current method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries to the machine precision.

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