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

Motivation

• Inexact boundary enforcement in PINN collocation and deep Ritz methods harms training and accuracy, especially on complex geometries.Boundary residuals create competing objectives whose weighting affects the realized solution and convergence.Source: §1 · pp. 2–3

Scope

• The approach is designed for affine, curved, and multiply connected geometries, extending PINNs toward meshfree simulation on exact geometries without mesh generation.The paper also applies R-functions to a Poisson problem on the 4D hypercube.Source: §1 · pp. 2, 4

Core approach

• The method constructs trial functions that eliminate boundary-loss terms while satisfying Dirichlet and mixed Dirichlet–Robin conditions a priori.This also ensures kinematic admissibility in deep Ritz formulations.Source: §1 · pp. 2, 4

Distance-field construction

• R-functions and mean value potential fields provide smooth approximate distance functions that vanish on boundaries and have unit inward normal derivative.Transfinite interpolation then supports the imposition of essential and Robin boundary conditions on complex domains.Source: §1 · p. 4

Results

• Numerical results across one-, two-, and four-dimensional boundary-value problems show that exact boundary imposition simplifies training and improves PINN accuracy and robustness.The examples include Poisson, plate bending, harmonic-coordinate, and Eikonal problems on convex, nonconvex, affine, and curved domains.Source: §1 · p. 4

Motivation and requirements

• ADFs provide closed-form alternatives to computationally expensive exact distance functions, whose limited smoothness can make them unsuitable for PDE trial functions.The exact distance solves the Eikonal equation and may be only continuous rather than continuously differentiable.Source: §2 · p. 4
• For essential boundary conditions, an ADF must vanish on ∂S, remain positive inside S, and have a nonvanishing boundary gradient.Smoothness is necessary because interior gradient discontinuities can make the Laplacian unbounded at collocation points for second-order problems.Source: §2 · pp. 4–5

Geometric notation

• The paper distinguishes the inward normal ν from the outward normal n, with n = −ν, and uses separate ADFs φ_i for individual boundary segments or curves.This notation supports the construction and analysis of distance functions on piecewise curved or linear boundaries.Source: §2 · p. 5

Normalization properties

• A normalized m-th-order ADF satisfies ∂φ/∂ν = 1 and ∂^kφ/∂ν^k = 0 for k = 2, 3, ..., m on regular boundary points.These conditions make the approximation match the exact distance function near the boundary, although finite-order ADFs deviate farther away.Source: §2 · p. 5

3. R-functions and Approximate Distance Functions

• R-functions compose signed functions through negation, union, and intersection operations, enabling approximate distance functions for arbitrarily complex boundaries.The construction associates positive, zero, and negative signs with interior, boundary, and exterior regions, respectively, and supports Boolean solid composition.Source: §3 · p. 5, Figure 1 · p. 6

3.1. Normalized functions for line segments and curves

• Normalized functions for segments, arcs, circles, and ellipses provide smooth approximate distances, while parametric curves generally require implicitization.The segment construction combines a signed line distance with a normalized trimming function, producing a C2 approximation away from the segment.Source: §3.1 · pp. 6–7, Figure 2 · p. 7, Figure 3 · p. 7

3.2. R-equivalence operation

• R-equivalence preserves distance normalization and associativity when joining boundary pieces, and increasing m improves exact-distance approximation but increases higher-order Laplacian undulations.For complex polygonal domains, the resulting contours are smooth and well-separated; the authors use m = 1 in most numerical simulations, despite prior use of m = 2.Source: §3.2 · p. 8

4. Generalized Mean Value Potentials and Lp-Distance Fields

• Generalized mean value potentials define smoothed approximate distance fields, with MVP denoting reciprocal mean value normalization and p = 1 a singular double-layer Lp-potential instance.The construction supports transfinite interpolation over curved boundaries and is presented separately for polygons and curved domains.Source: §4 · pp. 8, 10

4.1. Approximate distance fields on arbitrary planar polygons

• For arbitrary planar polygons, reciprocal mean value weights yield smooth ADFs, including nonconvex and nested polygons, normalized so the boundary normal derivative is 1.The resulting fields are illustrated for squares, L-shaped and nested domains, and a 291-vertex polygonalized map of Bhutan.Source: §4.1 · pp. 10–11, Figure 10 · p. 11, Figure 11 · p. 11

4.2. Approximate distance fields over curved domains

• For curved domains, generalized mean value potential fields produce approximate distance functions valid on convex and nonconvex geometries, smooth in the interior and compatible with polygonal limits.The p = 1 fields are shown for elliptical, annular, hypocycloid, and propeller-shaped domains; for a Ck boundary, φ is Ck on the boundary.Source: §4.2 · p. 12

5. Imposing Boundary Conditions in Deep Neural Networks

• The proposed trial function combines an approximate-distance-based boundary component with a neural-network approximation to satisfy essential and Robin boundary conditions exactly.The neural-network parameters remain unknown, while the boundary structure enforces the prescribed conditions.Source: §5 · pp. 12–13

5.1. Normalizing functions and solution structures

• Normalizing a function with respect to an m-th-order approximate distance function preserves its boundary value while forcing normal derivatives through order m to vanish.This yields solution structures that exactly satisfy prescribed boundary data and support Dirichlet, Neumann, and Robin constructions.Source: §5.1 · pp. 13–14

5.1.1. Solution structure for Dirichlet boundary condition

• For Dirichlet data u = g, the trial function uses a boundary extension plus the approximate distance function times any suitable numerical approximation, enforcing u = g on ∂Ω.Deep neural networks provide the numerical approximation in this paper.Source: §5.1.1 · p. 14

5.1.2. Solution structure for Neumann boundary condition

• Neumann solution structures are defined with respect to the outward normal and combine normalized boundary extensions with approximation functions to impose prescribed normal derivatives.This formulation departs from earlier R-function treatments by aligning the construction with the outward-normal direction used in boundary-value problems.Source: §5.1.2 · p. 14

5.1.3. Solution structure for Robin boundary condition

• The Robin ansatz is constructed analogously and is verified to satisfy ∂u/∂n + cu = h on ∂Ω.Robin conditions include the natural boundary condition as a particular case.Source: §5 · p. 12, §5.1.3 · p. 15

5.2. Imposing inhomogeneous essential boundary conditions

• For distinct boundary subsets with inhomogeneous essential data, transfinite interpolation blends boundary functions using partition-of-unity weights, while the resulting trial function preserves kinematic admissibility.The interpolation parameter µi controls whether only gi or both gi and ∂gi/∂n are interpolated.Source: §5.2 · p. 15

5.3. Imposing inhomogeneous essential and Robin boundary conditions

• For mixed essential and Robin boundaries, the method offers superposition for simple partitions and transfinite interpolation for general multiple-boundary configurations.The paper applies superposition to a one-dimensional mixed Dirichlet–Neumann problem and transfinite interpolation to a two-dimensional Dirichlet–Robin problem.Source: §5.3 · p. 15
• The mixed-boundary trial function combines separately constructed boundary structures, and the transfinite-interpolation formulation exactly imposes the mixed conditions.The construction uses distinct approximate distance functions for the boundary subsets and selects interpolation orders µ1 = 1 and µ2 = 2.Source: §5.3 · pp. 15–16

6.1. Feedforward neural network

• The approximation uses densely connected multilayer perceptrons, with neurons applying nonlinear activations across hidden layers and an output linear map.The paper considers deep networks as having at least two hidden layers and illustrates architectures with one, two, and four hidden layers.Source: §6 · p. 16, §6.1 · p. 17

6. Approximation of Trial Functions in a Deep Neural Network

• Trial functions are represented by nonlinear neural-network maps with unknown parameters obtained through an optimization problem, replacing standard linear combinations of known basis functions.Once optimized, the network parameters define the numerical solution to the boundary-value problem.Source: §6 · p. 16

6.1. Feedforward neural network

• The PINN approximates the boundary-value problem through a feedforward composition parameterized by layer weights and biases, combined with approximate distance functions in the trial function.The formulation covers differential equations in bounded domains with essential, natural, higher-order, or Robin boundary conditions.Source: §6.1 · p. 17

6.2. Backpropagation algorithm

• Network parameters are learned iteratively by backpropagation, using automatic differentiation to compute loss gradients and the Adam optimization algorithm.Adam adapts per-parameter learning rates using first and second gradient moments.Source: §6.2 · p. 17

7. Formulations

• The formulations exactly satisfy all boundary conditions in deep collocation, while deep Ritz satisfies the essential boundary conditions.These constructions target second- and fourth-order boundary-value problems.Source: §7 · p. 17

7.1. Deep collocation

• Deep collocation enforces the trial-function boundary conditions and minimizes mean squared PDE residuals at interior collocation points using Adam.The residual loss is measured with a discrete L2 norm over the domain.Source: §7.1 · p. 18
• Using a standard PINN trial function adds boundary residual terms, producing separate interior and boundary losses when only essential conditions are imposed.Exact boundary-condition trial functions remove these additional boundary residual contributions from the collocation loss.Source: §7.1 · p. 18

7.2. Deep Ritz

• Deep Ritz solves variationally structured second- and fourth-order problems by minimizing potential energy over a finite-dimensional PINN trial-function space.The formulation covers Poisson problems and Kirchhoff plate bending with the specified essential, mixed, or clamped conditions.Source: §7.2 · p. 18

7.2.1. Second-order problems

• For second-order Poisson problems, the Ritz formulation uses internal and external work functionals and enforces Dirichlet or mixed Dirichlet-Robin conditions through kinematically admissible trial functions.The resulting finite-dimensional problem is evaluated with Monte Carlo integration and solved by discrete nonlinear minimization.Source: §7.2.1 · pp. 18–19
• The second-order Ritz loss uses randomly or quasi-uniformly distributed, equally weighted domain points to approximate the potential energy functional.Stochastic gradient descent or a variant is used for the resulting minimization.Source: §7.2.1 · p. 19

7.2.2. Fourth-order problem

• For clamped Kirchhoff plate bending, both displacement and normal-derivative conditions must be satisfied, so the trial function incorporates an approximate distance function φ(x).The fourth-order Ritz formulation then uses the same Monte Carlo and loss-construction pattern as the Poisson problem.Source: §7.2.2 · pp. 19–20

Problem coverage

• Numerical solutions cover one-, two-, and four-dimensional boundary-value problems involving both second- and fourth-order equations.Deep collocation and deep Ritz methods are used throughout the numerical studies.Source: §8 · p. 20

Boundary-condition enforcement

• The approach exactly satisfies all collocation boundary conditions and essential Ritz conditions, making the trial function kinematically admissible.R-equivalence with m = 1 and mean value potential with p = 1 form the approximate distance functions unless otherwise stated.Source: §8 · p. 20

Software and differentiation

• The implementation uses JAX automatic differentiation to construct approximate distance functions, neural-network outputs, and derivatives for the governing loss functions.The homogeneous essential-boundary ansatz is formed as φ(x,y)*NN(params,x,y), and derivatives are obtained by repeated automatic differentiation.Source: §8 · pp. 20–21

Network architecture

• A frequently used architecture for problems in R^d is a two-hidden-layer d–N–N–1 network, typically with N = 50 and cubic ReLU hidden activations.The output layer uses a linear activation, while Ritz problems use ReLU or cubic ReLU except for one Gaussian-activation case.Source: §8 · pp. 20–21

Point generation

• Collocation points use uniform grids on simple domains and dmsh-generated triangular-mesh centroids on complex domains, with boundary points added when needed.All collocation points are processed in a single batch rather than sorted into bins.Source: §8 · pp. 21–22

9. Numerical Examples in One Dimension

• The one-dimensional study spans essential, mixed, discontinuous, singular, and point-load boundary-value problems, with comparisons against exact solutions and standard PINN.The authors emphasize demonstrating formulation benefits rather than optimizing hyperparameters or comparing against finite elements.Source: §9 · p. 22, §9.1 · p. 22, §9.1.5 · p. 26

9.1.1. Example 1

• Relative errors are O(10^-6) for the proposed method versus O(10^-5) for standard PINN [6] under identical larger-network hyperparameters.With a 1–50–50–1 network, 300 interior points, and 50,000 epochs, both methods improve substantially, but the proposed formulation remains more accurate.Source: §9.1.1 · p. 23, Figure 17 · p. 24

9.1.2. Example 2; 9.1.4. Example 4; 9.1.6. Example 6

• In the mixed-boundary comparison, the boundary-enforcing solution has a training loss about two orders smaller, a normalized absolute error one order smaller, and more accurate displacement and strain fields.Source: §9.1.2 · p. 24

9.1.3. Example 3

• Under sinusoidal forcing, the boundary-condition formulation shows good agreement with the exact solution for k = 5, while standard PINN at k = 1 fails to converge on 1–30–30–1 and 1–100–100–1 networks.For k = 1, a 1–100–100–100–1 network achieves error comparable to the boundary-condition formulation; accurate single-network solutions for high-frequency problems become infeasible due to spectral bias, motivating domain decomposition.Source: §9.1.3 · p. 24

9.1.5. Example 5

• For a point-load rod solved with the deep Ritz method, the proposed trial function reaches normalized absolute error O(10^-3) and closely matches the exact displacement.Displacement errors remain uniformly small, while strain errors increase near the load because the network is C2 but the exact solution is only C0 with discontinuous derivative.Source: §9.1.5 · p. 26, Figure 21 · p. 26

9.1.7. Example 7

• A single-hidden-layer Gaussian network captures an inhomogeneous Dirichlet solution with small displacement errors, although derivative errors remain appreciable for both collocation and Ritz methods.The example demonstrates that PINNs can represent variationally adaptive solutions with a Gaussian network, while more hidden neurons are expected to improve strain accuracy.Source: §9.1.7 · pp. 26–27, Figure 23 · p. 27

9.2. Longitudinal vibrations of a homogeneous elastic rod

• For longitudinal vibration eigenproblems, the proposed Ritz trial function uses only PDE and normalization losses, whereas standard PINN additionally penalizes boundary violations.The proposed and standard losses saturate near 1 and 10, respectively, in the lowest-mode calculation.Source: §9.2 · p. 28
• For longitudinal vibrations, the boundary-condition-enforcing solution has much closer agreement with the exact lowest mode, while the unconstrained solution remains less accurate after 10,000 epochs.The corresponding natural frequency is ω1 = π; the remaining mode-shape discrepancy is attributed to imperfect normalization.Source: §9.2 · p. 28, Figure 24 · p. 28

9.3. Advection-diffusion problem

• For α = 0, 5, 10, and 50, normalized absolute training errors are O(10−3) or less, with excellent agreement between numerical and exact solutions for all selected α.Source: §9.3 · p. 28

9.4. Euler-Bernoulli beam bending

• For Euler–Bernoulli beam bending, the boundary-condition-enforcing solution reaches loss O(10^-4), matches the exact solution, and yields accurate deflection and rotation fields.The unconstrained solution remains near loss 0.1 and has large field errors; improving it requires a boundary-to-interior loss-weight ratio of 10^4 for comparable accuracy.Source: §9.4 · p. 29, Figure 26 · p. 30

10. Numerical Examples in Two Dimensions

• The study compares exact-boundary PINNs with equally weighted standard PINNs across four two-dimensional problem types, using exact or finite-element references and deep collocation.The problems include steady heat conduction, harmonic coordinates, clamped Kirchhoff plate bending, and the Eikonal equation on polygonal and curved domains.Source: §10 · p. 30
• Approximate distance fields from R-equivalence and mean value potentials must have sufficient regularity because they enter the PINN trial function and its derivatives.Their regularity matters in both deep collocation and deep Ritz formulations.Source: §10 · p. 30

10.1. Laplacian of approximate distance fields

• On the unit square, both R-equivalence and mean-value-potential fields vanish on the boundary and are monotonic and concave inside, but their Laplacians are singular at polygon vertices.The fields and their first derivatives remain bounded, whereas their second derivatives become very large near vertices.Source: §10.1 · p. 30, Figure 27 · p. 31
• Vertex singularities can dominate Poisson collocation losses, whereas they do not affect Ritz solutions because the variational formulation uses only first derivatives.This distinction follows from the derivative order required by each formulation.Source: §10.1 · p. 30
• Restricting interior collocation points to [0.01, 0.99]^2 avoids points near square vertices, and both approximate-distance methods then perform well.The paper adopts this remedy rather than modifying the distance fields to bound their Laplacians.Source: §10.1 · pp. 30–31
• Exact distance functions are generally unsuitable for collocation when their medial-axis singularities lie inside the domain, though they remain usable when those singularities are outside it.The annulus and hollow-cylinder cases are cited as exceptions.Source: §10.1 · p. 31

10.2.1. Essential boundary conditions

• REQ- and MVP-based trial functions achieve errors an order of magnitude smaller than standard PINNs for homogeneous Dirichlet heat conduction.Although standard PINNs can attain comparable or lower training losses, their boundary errors remain large because the boundary-loss term is undervalued; exact enforcement avoids ad-hoc loss-weight tuning.Source: §10.2.1 · pp. 32–33, Figure 28 · p. 32

10.2.2. Example 2

• Transfinite interpolation with an approximate distance function exactly satisfies nonzero Dirichlet data on separate unit-square boundary edges, with REQ errors almost two orders of magnitude below standard PINNs.REQ is also almost an order of magnitude more accurate than MVP, while both geometry-aware schemes have precisely zero boundary error.Source: §10.2.2 · pp. 33–34
• The Ritz formulation remains well-defined near boundaries and reaches O(10^-2) error in fewer than 2,000 epochs, outperforming MVP collocation but not REQ collocation.Sampling close to the boundary prevents trivial constant solutions by exposing their large boundary gradients in the loss.Source: §10.2.2 · p. 35, Figure 34 · p. 36

10.2.3. Curved domain

• On an annulus, a 2–50–50–1 REQ network converges acceptably, whereas standard PINN requires 2–150–150–1 yet retains errors two orders of magnitude larger.The standard PINN also fails to satisfy the curved-domain boundary conditions, with especially large errors on the boundaries.Source: §10.2.3 · pp. 36–37, Figure 36 · p. 37
• For mixed annulus conditions, transfinite interpolation produces a trial function satisfying both Dirichlet and Robin boundaries, yielding a solution within 1 percent of the exact solution.The retained Dirichlet condition is u = 1 on Γ1, while Γ2 uses ∂u/∂n + u = 2 + 4 ln 4.Source: §10.2.3 · pp. 37–38
• The method is evaluated on a Laplace problem in an annulus with mixed boundary conditions.The figure reports the numerical solution obtained with ˜ubc nn(x; θ).Source: Figure 37 · p. 38
• Training loss and normalized absolute errors are tracked as functions of epochs.Source: Figure 37 · p. 38
• Surface plots visualize the computed solution for the curved-domain problem.Source: Figure 37 · p. 38

10.3. Generalized barycentric coordinates over polygons

• On polygons, each harmonic coordinate solves a Laplace equation with piecewise affine hat-function boundary data and remains positive in the interior.Positivity follows from the maximum principle; boundary data are assembled through transfinite interpolation.Source: §10.3 · p. 38

10.3.1. Harmonic coordinates on a square

• The square benchmark has an exact solution, u(x, y) = (1 − x)(1 − y), and the Ritz method is natural because harmonic coordinates minimize Dirichlet energy.The same polygonal framework uses transfinite interpolation to impose all edge boundary data simultaneously.Source: §10.3 · p. 38, §10.3.1 · p. 39
• On the square, the boundary-conforming network exactly satisfies essential conditions and has far smaller errors than the unconstrained network.The unconstrained formulation becomes comparable only when its boundary-loss weight is set to w = 10^-3, making that term 1,000 times more important than the PDE loss.Source: §10.3.1 · p. 39

10.3.2. Harmonic coordinates on an L-shaped polygon

• On the L-shaped polygon, REQ and MVP approximations achieve about 3% maximum error away from the reentrant corner and about 20% near its singularity.Normalized absolute errors are O(10^-2) for the vertex at the origin and O(10^-1) for the reentrant-corner vertex, with finite-element references used for comparison.Source: §10.3.2 · pp. 39–40

Problem setup

• The problem solves the biharmonic equation ∇4u = 1 on the unit disk, where both u and ∂u/∂n are prescribed on the boundary.The boundary conditions motivate a solution structure different from those used in earlier problems.Source: §10.4 · p. 40

Boundary-condition construction

• The trial function ũ_bc_nn(x; θ) = φ^2ũ(x; θ) satisfies both clamped boundary conditions by using a distance function to the unit circle.The exact distance function has derivative singularities at the origin, so an approximate distance function is used; the resulting φ is a bivariate polynomial.Source: §10.4 · pp. 40–42

Implementation

• The network architecture is 2–50–50–1, and cubic ReLU is used for collocation and Ritz, although standard PINN collocation cannot use cubic ReLU for this biharmonic problem.The constructed trial function contains additional terms and is therefore generally not biharmonic.Source: §10.4 · p. 42

Numerical results

• Both Ritz and collocation methods achieve unnormalized errors of O(10^-5) for the clamped circular Kirchhoff plate.The computations use 2800 interior points and the trial function ũ_bc_nn(x; θ).Source: §10.4 · p. 42

Method and setup

• The method solves the Eikonal equation with f = 1 using collocation and ADFs built from R-equivalence and generalized mean-value potentials, without upwind causality restrictions.The solution represents signed distance, while the constructed trial functions impose u = 0 exactly on the interface.Source: §10.5 · p. 43

Square interface

• For the square, MVP and REQ enforce the boundary exactly and achieve L∞ errors of 0.026 and 0.03, versus 1.34 for the standard PINN.Although the standard PINN attains the lowest training loss, it has the highest pointwise error; MVP is slightly more accurate throughout the domain.Source: §10.5 · p. 43, Figure 43 · p. 43

Ellipse interface

• For an ellipse, errors remain below 1 percent away from the center, where the exact distance is C0 but the neural approximation is C2 smooth.The standard PINN results were very poor and were omitted; REQ and MVP use a 2–50–50–1 architecture.Source: §10.5 · pp. 43–44

Bhutan-map interface

• For the polygonalized Bhutan map, the normalized error is O(10^-1) at 20,000 epochs and the maximum error is about 4 percent.REQ and MVP are evaluated with a 2–50–50–1 network, while standard PINN results are omitted because they are very poor.Source: §10.5 · p. 44

Problem setup

• The problem uses a Poisson equation on a four-dimensional hypercube with homogeneous Dirichlet conditions to demonstrate exact boundary enforcement in higher dimensions.High-dimensional PINN problems are challenging because meshing is difficult and the curse of dimensionality applies.Source: §11 · p. 44

Trial-function construction

• The REQ trial function replaces the product method’s poorly scaled factor, which decays as O(x^8), with a higher-dimensional distance function scaling as x_i^2 in each coordinate.REQ is formed from R-equivalence operations on strip distance functions and generalizes to the hypercube in R^d.Source: §11 · p. 45

Results

• With 5,000 randomly generated interior points for training and a separate evaluation set of 5,000 interior points, REQ reaches about 1% normalized error, while the product method does not converge and retains O(1) error despite O(10^-5) PDE loss.Source: §11 · pp. 45–46

• The method constructs approximate distance fields with R-functions and mean value potential fields, embedding them into PINN and Ritz formulations for exact boundary satisfaction.It enforces all boundary conditions in collocation methods and essential conditions in Ritz methods over affine and curved domains.Source: §12 · p. 46
• Exact boundary-condition enforcement makes training loss coincide with PDE loss, simplifying optimization and producing more accurate solutions than standard weighted-loss PINN formulations.Fixed loss weights do not reliably indicate approximation accuracy because their appropriate values depend on the PDE, boundary conditions, and training.Source: §12 · pp. 46–47
• R-equivalence and mean value distance fields achieve the same accuracy order, while their suitability depends on boundary complexity and whether conditions vary across boundary segments.Mean value potentials are more efficient for polygons with many edges, whereas R-equivalence is required for distinct conditions on different boundary segments.Source: §12 · p. 47
• The formulation enables meshfree analysis without domain discretization and supports exact modeling of complex affine and curved geometries, providing a pathway toward isogeometric analysis.Transfinite inverse-distance interpolation combines with approximate distance functions to impose boundary conditions on complex two-dimensional domains.Source: §12 · p. 47
• R-equivalence extends to higher dimensions without the curse of dimensionality, demonstrated by an accurate PINN solution for Poisson’s equation on a 4D hypercube.Future work includes complex three-dimensional geometries and a deep Petrov-Galerkin domain-decomposition method.Source: §12 · p. 47