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Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

N. Sukumar, Ankit Srivastava

arXiv:2104.08426v2math.NAcs.NE

TL;DR

PINNs can suffer from inaccurate boundary enforcement, particularly on complex geometries. This paper constructs geometry-aware trial functions from approximate distance fields so boundary conditions are satisfied a priori, and numerical studies report more accurate solutions with simpler training, including a 4D hypercube example. The approach remains bounded by smoothness requirements for collocation and implicitization requirements for general parametric curves.

  • Problem

    Inexact boundary imposition adversely affects PINN training and accuracy, with the difficulty becoming more acute on complex geometries.

  • Method

    The method uses R-functions, mean value potential fields, and transfinite interpolation to construct approximate distance fields within trial functions that satisfy boundary conditions a priori.

  • Results

    Numerical examples report that exact boundary imposition simplifies training and leads to more accurate solutions across benchmark one- and two-dimensional problems, with an accurate Poisson solution also obtained on a 4D hypercube.

  • Takeaways & Limitations

    The study provides a meshfree route to solving PDEs on complex geometries without domain discretization while exactly satisfying the imposed boundary conditions.

  • Takeaways & Limitations

    Collocation generally requires smooth approximate distance functions, and parametric curves generally require implicitization to construct them.

Abstract

from arXiv · show

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artifical neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct $φ$, an approximate distance function to the boundary of a domain. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as $φ$ multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. We present numerical solutions for linear and nonlinear boundary-value problems over domains with affine and curved boundaries. Benchmark problems in 1D for linear elasticity, advection-diffusion, and beam bending; and in 2D for the Poisson equation, biharmonic equation, and the nonlinear Eikonal equation are considered. The approach extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the 4D hypercube. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

1. Introduction

The introduction identifies inaccurate boundary enforcement as a central PINN difficulty and presents geometry-aware trial functions that impose boundary conditions a priori. The approach uses approximate distance functions built with R-functions and mean value potential fields, supporting collocation and Ritz formulations.

  • The introduction positions PINNs alongside meshfree and neural-network approaches for solving PDEs, including collocation, Ritz, radial-basis, and Gaussian-network methods.
  • Inexact boundary conditions adversely affect PINN training and accuracy, especially on complex geometries.
  • The method constructs trial functions that satisfy all collocation boundary conditions a priori and enforce kinematic admissibility in deep Ritz methods.This removes boundary terms from the loss function.
  • Approximate distance functions are constructed using R-functions and mean value potential fields, which remain smooth in the domain and have unit inward normal derivative on the boundary.

2. Distance Functions and their Properties

Distance functions provide implicit geometric representations, but trial functions require boundary normalization and sufficient interior smoothness. Approximate distance functions therefore must reproduce boundary behavior while avoiding derivative discontinuities that impede collocation.

  • The exact distance function gives the shortest distance to the boundary and is identically zero on the boundary.
  • For complete essential boundary conditions, an approximate distance function must vanish on the boundary, remain positive inside, and have a nonvanishing boundary gradient.
  • The inward and outward boundary normals satisfy n = −ν, with ν used for distance-function theory and n for Neumann or Robin conditions.
  • An m-th-order normalized distance function has unit first normal derivative and vanishing second- through m-th-order normal derivatives.These conditions are expressed as ∂φ/∂ν = 1 and ∂^kφ/∂ν^k = 0 for k = 2, 3, . . . , m.
  • Finite-order normalized functions match exact distance only near the boundary and deviate farther away.

3. R-functions and Approximate Distance Functions

R-functions combine geometric predicates into approximate distance fields for complex boundaries, while trimming and normalization produce smooth fields for line segments and curves. R-equivalence provides an associative way to join component fields while preserving boundary normalization.

  • R-functions: R-functions combine signed component functions through Boolean operations such as union, intersection, complement, and equivalence.Their sign is determined solely by the signs of their arguments, allowing set-theoretic geometry to be represented algebraically.
  • R-functions: R-disjunction and R-conjunction represent union and intersection, and smoothing makes the selected operations C^s-continuous away from simultaneous zeros.
  • Normalized functions for line segments and curves: For a line segment, the construction combines a signed distance to its supporting line with a normalized trimming function to form an approximate distance field.The resulting field is C^2 away from the segment and avoids the derivative discontinuity of |t|.
  • Normalized functions for line segments and curves: The same construction is illustrated for quarter-circular arcs, circles, and ellipses using approximate distance functions normalized to first order.
  • Normalized functions for line segments and curves: For parametric curves such as Bézier and NURBS curves, constructing approximate distance functions generally requires curve implicitization.
  • R-equivalence operation: R-equivalence joins component distance fields while preserving normalization and associativity, unlike the alternative R-conjunction joining operation.
  • R-equivalence operation: Increasing the normalization order improves approximation away from segments and boundary normalization but produces higher-order fields with greater Laplacian undulations.The numerical simulations adopt m = 1 in most cases.

4. Generalized Mean Value Potentials and Lp-Distance Fields

The paper constructs smooth approximate distance fields over polygonal and curved domains using generalized mean value potentials, with behavior tied to normalization and potential-field parameters.

  • 4. Generalized Mean Value Potentials and Lp-Distance Fields: Generalized barycentric coordinates extend barycentric coordinates from simplices to polygons and polyhedra, supporting interpolation over convex and curved domains.Mean value coordinates are valid on arbitrary planar polygons, including nested polygons, and are smooth in the interior.
  • 4. Generalized Mean Value Potentials and Lp-Distance Fields: Increasing the R-equivalence normalization parameter m makes the approximate distance field approach the exact distance in the interior of a curved triangle.For two line segments, the paper compares R-conjunction and R-equivalence compositions at different normalization orders.
  • 4.1. Approximate distance fields on arbitrary planar polygons: For polygonal domains, the mean value potential construction yields approximate distance fields for triangles, squares, hexagons, L-shaped domains, nested polygons, and a 291-vertex polygonalized map of Bhutan.The resulting contour fields are described as smooth in the interior and monotonic over the domain for the Bhutan example.
  • 4. Generalized Mean Value Potentials and Lp-Distance Fields: The reciprocal of a mean value normalization function provides a smoothed approximate distance function to boundaries of polygons and curved domains.This construction is called the mean value potential (MVP) method and corresponds to the p = 1 case of generalized Lp-potential fields.
  • 4.2. Approximate distance fields over curved domains: For curved domains, generalized mean value potentials use a boundary integral whose p = 1 specialization recovers the polygonal mean value potential construction.The paper illustrates the resulting fields on elliptical disks, annuli, hypocycloids, and propeller-shaped domains.

5. Imposing Boundary Conditions in Deep Neural Networks

The paper embeds approximate distance functions and transfinite interpolation into neural-network trial functions so boundary conditions are satisfied a priori on complex geometries.

  • 5. Imposing Boundary Conditions in Deep Neural Networks: The PINN trial function is constructed to exactly satisfy essential and Robin boundary conditions before optimization.The neural-network approximation supplies the unknown remainder term within the boundary-compatible solution structure.
  • 5.2. Solution structure for Neumann boundary condition: Normalizing the approximate distance function to order m yields a solution structure whose first m normal derivatives vanish on the boundary.The construction uses inward and outward normal directions consistently for distance-function theory and Neumann or Robin conditions.
  • 5.1. Solution structure for Dirichlet boundary condition: For homogeneous Dirichlet conditions, multiplying the neural-network approximation by an approximate distance function makes the trial function vanish on the boundary.For inhomogeneous Dirichlet conditions, the solution structure combines boundary data with the distance function and a numerical approximation.
  • 5.1.3. Solution structure for Robin boundary condition: Robin boundary conditions are imposed by a dedicated ansatz that is verified to satisfy ∂u/∂n + cu = h on the boundary.The paper presents this as a general solution structure for Robin data.
  • 5.3. Imposing inhomogeneous essential and Robin boundary conditions: For multiple boundary subsets, transfinite interpolation combines prescribed boundary functions while distance fields enforce the corresponding boundary behavior.The interpolation parameter μ_i controls whether only boundary values or also normal derivatives are interpolated.
  • 5.3. Imposing inhomogeneous essential and Robin boundary conditions: For disjoint Dirichlet and Robin boundaries, the paper describes both superposition and transfinite-interpolation approaches, applying them to separate example problems.The first approach is used for a one-dimensional mixed Dirichlet-Neumann problem, while the second is used for a two-dimensional Dirichlet-Robin problem.

6. Approximation of Trial Functions in a Deep Neural Network

The paper represents PINN trial functions with densely connected neural networks whose trainable parameters are optimized to solve boundary-value problems. The formulation defines the governing problem, boundary operators, and network composition used for the approximation.

  • Network architecture: The paper uses multilayer perceptrons, in which densely connected neurons apply nonlinear activation functions across multiple hidden layers.The network maps an input point x ∈ R^d to a scalar approximation of u(x).
  • Trial-function representation: Unlike standard basis expansions, the PINN trial function is a nonlinear map with unknown parameters determined by minimizing an objective function.The optimization is generally, though not necessarily, a least-squares problem.
  • Problem statement: The boundary-value problem combines a differential equation in Ω with essential, natural, or higher-order boundary conditions on disjoint boundary subsets.For second-order problems, the higher-order boundary condition is a Robin condition.
  • Network parameterization: The neural-network approximation is parameterized by weights and biases and can be written as a composition of hidden-layer maps followed by a linear output map.The unknown parameter vector is θ := {W, b}.
  • Approximation properties: A sufficiently rich multilayer network can approximate any L2 function to arbitrary accuracy, but practical realization depends on architecture, activation, and optimization.The relevant choices include network width and depth, activation function, and the computational optimization algorithm.
  • Optimization: Network parameters are obtained iteratively through backpropagation, with automatic differentiation used to compute loss-function gradients efficiently.The implementation uses the Adam backpropagation algorithm.

7. Formulations

The formulations use geometry-aware trial functions to enforce boundary conditions before optimization, then solve the governing equation through deep collocation or deep Ritz minimization. The section covers second- and fourth-order problems, including mixed and clamped conditions.

  • 7. Formulations: The formulation exactly satisfies all boundary conditions in deep collocation, while deep Ritz enforces the essential boundary conditions.This distinction separates residual-based enforcement from kinematic admissibility in the variational method.
  • 7.1. Deep collocation: Deep collocation requires the boundary-condition-compatible trial function to satisfy the governing equation at interior collocation points.The residual at those points is minimized through a mean squared residual objective.
  • 7.1. Deep collocation: With a standard PINN trial function, the collocation loss includes separate interior PDE and boundary-loss contributions.The boundary term measures the discrepancy on boundary collocation points.
  • 7.2. Deep Ritz: The deep Ritz method applies a variational principle to finite-dimensional trial functions for second-order Poisson and fourth-order plate-bending problems.The considered Poisson cases include essential and mixed boundary conditions, while plate bending uses clamped conditions.
  • 7.2.1. Second-order problems: For Poisson problems with Dirichlet conditions, the variational formulation uses internal and external work functionals over an appropriate Sobolev space.The trial and test functions enter the symmetric bilinear and linear functionals.
  • 7.2.1. Second-order problems: Mixed Poisson conditions combine prescribed values on Γu with a Robin condition involving the outward normal derivative, coefficient c, and boundary data h on Γn.The boundary subsets are disjoint and together form the full boundary.
  • 7.2.1. Second-order problems: The deep Ritz trial function is chosen to satisfy essential conditions before minimization, making the finite-dimensional approximation kinematically admissible.Potential energy is evaluated using Monte Carlo integration with randomly or quasi-uniformly distributed, equally weighted points.
  • 7.2.2. Fourth-order problem: For clamped Kirchhoff plate bending, the trial function must satisfy both zero displacement and zero normal derivative on the boundary.The plate formulation uses a fourth-order strong problem with unit flexural rigidity and a variational principle based on H2(Ω).

8. Numerical Implementation

The numerical implementation combines neural-network ansätze with approximate distance functions, automatic differentiation, and interior point sampling. The code examples use R-equivalence distance functions and cubic ReLU networks to construct boundary-compatible approximations.

  • Numerical scope: The paper evaluates deep-neural-network solutions for one-, two-, and four-dimensional boundary-value problems of second and fourth order.The numerical section uses the boundary-condition-compatible trial function in these examples.
  • Boundary-compatible ansatz: For polygonal domains, the trial function is constructed as the product of an approximate distance function φ(x, y) and the neural-network output.This product satisfies homogeneous essential boundary conditions on the polygon boundary.
  • Method comparison: The implementation distinguishes the proposed loss and solution from those of standard PINNs using L_bc nn(θ), L_nn(θ), u_bc nn(x), and u_nn(x).The proposed approach is used with deep collocation and deep Ritz methods.
  • Distance functions: R-equivalence with m = 1 and mean value potential with p = 1 are the default constructions for approximate distance functions unless otherwise stated.The paper refers to the corresponding expressions as REQ and MVP constructions.
  • R-equivalence implementation: The R-equivalence code combines line-segment distance functions through reciprocal powers to return an approximate distance from a point to a polygon.The polygon is represented by a NumPy array containing the coordinates of its line segments.
  • Neural-network implementation: The implementation uses cubic ReLU, an instance of the RePU activation family, in hidden layers and a linear activation in the output layer.The optimizable network parameters are weights and biases.
  • Differentiation and losses: Automatic differentiation forms first- and second-order derivatives of the boundary-compatible trial function for constructing interior Ritz or collocation losses.The examples include gradients with respect to x and y and mixed second derivatives.
  • Point generation: Uniform grids provide points for square-domain, Eikonal, and four-dimensional hypercube problems, while dmsh triangular meshes supply centroids for complex domains.Figure 15 presents representative meshes and corresponding collocation points.

9. Numerical Examples in One Dimension

The one-dimensional examples compare standard PINN with geometry-aware trial functions that exactly satisfy boundary conditions. Across elasticity, advection-diffusion, Ritz, vibration, and beam-bending tests, the boundary-condition-aware formulation generally achieves more accurate solutions and lower losses.

  • Method: The formulation uses transfinite interpolation for g(x), a neural network approximation, and an approximate distance function φ(x) to construct the trial function.The approximate distance function may be selected from φA(x), φB(x), or φC(x).
  • Example 1: In the first Dirichlet rod example, relative errors were O(10^-5) for standard PINN and O(10^-6) for the proposed approach.Both methods improved with a larger network, more interior points, and longer training, while the proposed approach remained more accurate under matched hyperparameters.
  • Example 2: Using different approximate distance functions, the proposed solution had training loss about two orders lower and normalized absolute error one order lower than standard PINN.Its displacement and strain fields were also markedly more accurate over the full interval.
  • Examples 3–5: For a problem with a singular forcing profile, the proposed approach maintained accurate displacement and strain fields while standard PINN produced appreciable errors.The paper attributes the standard PINN error to poorly approximated boundary conditions, whereas the proposed trial function satisfies them exactly.
  • Example 4: At 10,000 epochs, the proposed method reached training loss O(10^-3) and normalized absolute error O(10^-5), while standard PINN stagnated near 0.1 with error O(10^-4).When both boundary conditions were homogeneous, standard PINN accuracy was comparable to the proposed approach.

10. Numerical Examples in Two Dimensions

The two-dimensional examples examine geometry-aware PINN formulations across polygonal and curved domains, while highlighting how approximate distance-field regularity affects collocation.

  • The study compares exact-boundary PINN formulations with standard PINN across heat conduction, harmonic coordinates, plate bending, and Eikonal problems.Solutions are compared with exact or reference finite element solutions and deep collocation.
  • Approximate distance functions must vanish on the boundary and remain sufficiently smooth in the interior for collocation methods using differential operators.For second-order problems, interior gradient discontinuities can make the Laplacian unbounded at collocation points.
  • R-equivalence and mean value potential fields produce distance functions that vanish on the square boundary and are monotonic inside the domain.Their Laplacians are singular at polygon vertices and can become very large nearby.
  • Exact distance functions can be unsuitable when their medial-axis singularities lie inside the computational domain, motivating smooth approximate distance functions.Annular domains are identified as an exception where the medial axis is excluded from the computational region.

10.2. Steady-state heat conduction

The heat-conduction examples show that distance-function trial functions satisfy essential boundary conditions exactly and substantially reduce prediction errors relative to standard PINN formulations.

  • 10.2.1. Essential boundary conditions: The geometry-aware ansatz multiplies the neural-network approximation by a boundary-zero function, thereby satisfying homogeneous essential conditions automatically.REQ and MVP constructions generalize this mechanism to complex domains.
  • 10.2.1. Essential boundary conditions: REQ- and MVP-based schemes deliver an order of magnitude more accurate heat-conduction solutions than the unconstrained standard PINN.This occurs despite the geometry-aware formulation sometimes having a larger training loss.
  • 10.2.1. Essential boundary conditions: Standard PINN errors are especially large near the boundary because the PDE loss can dominate and undervalue the boundary-loss term.Increasing the boundary-loss weight can improve standard PINN results, but this is described as ad hoc.
  • 10.2.1. Essential boundary conditions: The relative boundary-loss weight has a problem-dependent sweet spot, whereas the exact-boundary approach requires no such tuning.For the reported comparison, w = 0.1 weights the boundary loss nine times more than the PDE loss and performs better than w = 0.5 or 0.9.
  • 10.2.2. Example 2: For the reported square example, REQ errors are almost an order of magnitude below MVP errors and almost two orders below standard PINN errors.Boundary errors for both REQ and MVP are precisely zero, while standard PINN has large boundary errors.
  • 10.2.2. Example 2: The Ritz formulation with REQ reduces error to O(10^-2) in fewer than 2,000 epochs, with errors smaller than MVP collocation but larger than REQ collocation.The computation uses 5,000 interior points sampled near the boundary.
  • 10.2.3. Curved domain: On an annulus, the REQ ansatz exactly matches the imposed boundary data, and the numerical solution remains within 1 percent of the exact solution.The example uses transfinite interpolation to construct composite boundary data.

10.3. Generalized barycentric coordinates over polygons

The polygon examples use harmonic coordinates and transfinite interpolation on square and L-shaped domains, including a reentrant-corner singularity that increases approximation error.

  • 10.3. Generalized barycentric coordinates over polygons: Harmonic coordinates are generalized barycentric coordinates obtained by solving Laplace problems with piecewise affine vertex-based Dirichlet data.Each coordinate is nonnegative and positive in the polygon interior.
  • 10.3. Generalized barycentric coordinates over polygons: Transfinite interpolation assembles the boundary data used to compute harmonic coordinates on square and L-shaped polygons.The reported network architecture for both examples is 2–50–50–1.
  • 10.3. Generalized barycentric coordinates over polygons: On a square, harmonic coordinates coincide with bilinear finite element shape functions, and the vertex-associated solution is u(x,y) = (1-x)(1-y).The Ritz formulation is natural because harmonic coordinates minimize Dirichlet energy.
  • 10.3. Generalized barycentric coordinates over polygons: The exact-boundary harmonic-coordinate ansatz satisfies essential boundary conditions and has far smaller errors than the unconstrained formulation.Standard PINN performance becomes comparable only after tuning the boundary-loss weight to w = 10^-3.
  • 10.3. Generalized barycentric coordinates over polygons: For the vertex at (1/2,1/2), maximum error near the reentrant-corner singularity reaches about 20 percent, a 10-fold increase over the other L-shaped case.The corresponding normalized absolute error is O(10^-1).

10.4. Clamped circular Kirchho

The clamped circular plate example uses a squared approximate distance function to impose both boundary conditions and evaluates collocation and Ritz solutions for a fourth-order PDE.

  • The circular plate problem specifies both displacement and normal-derivative boundary conditions, requiring a solution structure different from earlier examples.The construction begins with a distance function to the circular boundary.
  • An approximate polynomial distance function is used because the exact circular distance has derivative singularities at the origin.The ansatz is ˜u_bc_nn(x; θ) = φ^2 ˜u(x; θ), enforcing both essential conditions.
  • The same geometry-aware trial function supports both collocation and Ritz solutions for the clamped circular plate.Both methods use 2,800 interior points and a 2–50–50–1 network.
  • The Ritz and collocation solutions achieve unnormalized errors of O(10^-5) after 10,000 epochs.The reported collocation accuracy is consistent with the cited prior result.

10.5. Eikonal equation

The Eikonal experiments evaluate PINN distance-function solutions on square, ellipse, and polygonal geometries. Exact-boundary trial functions achieve lower pointwise errors than the standard PINN, including on nonsmooth and complex interfaces.

  • Problem setup: With f = 1, the Eikonal solution represents the signed distance to the boundary, and the study computes it using PINN on affine and curved interfaces.The square benchmark embeds the interface in the biunit square and imposes u = 0 on the smaller square boundary.
  • Square: For the square, REQ and MVP trial functions satisfy the boundary condition exactly, while standard PINN has the largest domain-wide error.The L∞ errors are 0.03 for REQ, 0.026 for MVP, and 1.34 for the standard PINN.
  • Square: MVP is slightly more accurate than REQ across the square domain despite the standard PINN attaining the lowest training loss among the three schemes.This illustrates that loss magnitude alone does not identify the most accurate solution when boundary satisfaction differs.
  • Ellipse: For the ellipse, errors are below 1 percent away from the center, where larger errors concentrate because the exact distance function has derivative discontinuities.The exact distance is C0 at the center, whereas the numerical solution is C2 smooth.
  • Bhutan: For the polygonalized map of Bhutan, the maximum error is about 4 percent, while the standard trial-function results are omitted because they are very poor.At 20,000 epochs, the absolute normalized training error is O(10^-1).

11. Poisson Problem over the Four-Dimensional Hypercube

The four-dimensional hypercube experiment compares product and R-equivalence trial functions for a Poisson problem with homogeneous Dirichlet conditions. R-equivalence converges accurately, whereas the product construction fails despite a smaller PDE loss.

  • Problem setup: The study solves a Poisson problem with homogeneous Dirichlet boundary conditions on a four-dimensional hypercube to demonstrate higher-dimensional applicability.The experiment uses a 4–100–100–1 network and evaluates normalized error on separate interior points.
  • Trial functions: The product method creates very small interior factors, scaling as O(x^8), which forces network parameters to compensate for highly nonlinear behavior during training.R-equivalence is introduced as a better-behaved alternative that extends additively to higher dimensions.
  • Trial functions: R-equivalence constructs the approximate distance function by composing strip-based R-functions associated with the hypercube’s bounding hyperplanes.Each strip function is normalized to order 1 before R-equivalence composition.
  • Results: R-equivalence reaches error levels of about 1 percent, whereas the product method does not converge under the reported training setup.Both methods use 5,000 randomly generated interior training points and a separate set of 5,000 points for error evaluation.
  • Results: The REQ solution has O(10^-2) error, while the product method has O(1) error despite a much smaller PDE loss of O(10^-5).The comparison demonstrates that PDE loss magnitude can disagree sharply with solution accuracy in this boundary-enforcing setting.

12. Conclusions

The paper introduces distance-field-based geometry-aware PINN trial functions that impose boundary conditions a priori. Numerical studies report simpler training and accurate solutions, including a four-dimensional Poisson demonstration, while leaving complex 3D extensions for future work.

  • Contributions: The method uses R-functions and transfinite interpolation with approximate distance functions to impose necessary boundary conditions a priori in PINNs.The approach targets complex affine, curved, and multiply connected geometries.
  • Findings: Exact boundary enforcement leaves only the interior residual contribution in the loss, simplifying training and improving numerical accuracy in the reported verification tests.The conclusions contrast this with standard PINNs, where fixed loss weights can make final loss magnitude a poor accuracy indicator.
  • Comparison of ADFs: The R-equivalence and mean-value-potential ADFs generally achieve the same order of accuracy, with R-equivalence usually having the smaller constant except for the Eikonal equation.Mean value potentials are more efficient for polygons with many edges, while R-equivalence supports distinct boundary conditions on separate segments.
  • Scope: The formulation supports meshfree PDE analysis without domain discretization on complex two-dimensional geometries and extends to higher dimensions through additive R-equivalence composition.An accurate PINN solution is reported for a Poisson problem on the four-dimensional hypercube.
  • Future work: Extension to complex three-dimensional geometries and development of a deep Petrov–Galerkin domain-decomposition method remain planned topics.
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