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$\Delta$-PINNs: physics-informed neural networks on complex geometries

Francisco Sahli Costabal, Simone Pezzuto, Paris Perdikaris

arXiv:2209.03984v2cs.LG

TL;DR

PINNs are most often applied to simple geometries, while complex domains lack a clear way to convey their topology to the network. This paper encodes geometry with Laplace-Beltrami eigenfunctions and evaluates the approach on varied geometries and PDE problems. The authors report better performance than traditional PINNs, especially when Euclidean proximity differs from intrinsic distance.

  • Problem

    Most PINN applications use simple geometries, and there is no clear way to inform the network about domain topology.

    Source: §1 · p. 1

  • Method

    The method uses Laplace-Beltrami eigenfunctions as positional encodings and finite elements to approximate eigenfunctions and PDE operators.

    Source: §1 · p. 2

  • Results

    The method outperforms traditional PINNs, especially when points close in Euclidean distance are far apart intrinsically.

    Source: §4 · p. 16

  • Takeaways & Limitations

    Eigenfunction encodings provide a compact representation of geometric shape information for PINNs.

    Source: §1 · p. 2

    The study requires a mesh to compute eigenfunctions and PDE operators, and more sophisticated finite elements may be needed for some operators.

    Source: §4 · p. 16

Abstract

from arXiv · show

Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations. Despite recent progress on expanding the class of problems that can be tackled by PINNs, most of existing use-cases involve simple geometric domains. To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved. In this work, we propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator. This technique allows to create an input space for the neural network that represents the geometry of a given object. We approximate the eigenfunctions as well as the operators involved in the partial differential equations with finite elements. We extensively test and compare the proposed methodology against traditional PINNs in complex shapes, such as a coil, a heat sink and a bunny, with different physics, such as the Eikonal equation and heat transfer. We also study the sensitivity of our method to the number of eigenfunctions used, as well as the discretization used for the eigenfunctions and the underlying operators. Our results show excellent agreement with the ground truth data in cases where traditional PINNs fail to produce a meaningful solution. We envision this new technique will expand the effectiveness of PINNs to more realistic applications.

1. Motivation

PINNs have advanced across physical domains, but most work still uses simple geometries, limiting their relevance to problems with complex shapes and topology. The paper proposes encoding geometry with Laplace-Beltrami eigenfunctions and tests this approach across several geometries and physical problems.

1. Motivation

  • Most PINN applications still use simple geometric domains, although real-world objects may have complicated shapes and topologies.Prior approaches include signed-distance boundaries, domain decomposition, and coordinate mappings, but the cited examples concern 2D shapes.Source: §1 · pp. 1–2
  • Euclidean coordinates can obscure intrinsic distances on complex domains, where nearby embedded points may be far apart along the surface.Existing methods for 3D surfaces work well when Euclidean and intrinsic distances are similar, but that condition does not hold in all applications.Source: §1 · p. 2
  • The proposed positional encoding uses Laplace-Beltrami eigenfunctions to represent geometry and preserve proximity between points in the geometry.The network receives eigenfunction values at each point; the functions can be approximated with finite elements, as can PDE operators such as the gradient and Laplacian.Source: §1 · p. 2
  • The paper tests the method on a coil, heat sink, and bunny, including Eikonal and heat-transfer problems.The planned study also includes Poisson and hyperelastic problems, geodesic-distance learning, comparisons with other PINNs, and mesh-size scaling.Source: §1 · p. 2

2. Methods

The methods formulate PINNs using data, PDE, and boundary-condition constraints, then replace Euclidean coordinates with Laplace-Beltrami eigenfunction features. Finite elements provide both the geometry-dependent eigenfunctions and the differential operators needed to evaluate PDE constraints.

2.1. Physics-informed neural networks

  • The PINN formulation fits partial observations while penalizing PDE residuals and boundary-condition violations, but Euclidean inputs do not encode complex geometry.PDE residuals may be evaluated at randomly generated or predefined interior locations, and Neumann conditions at boundary points.Source: §2.1 · pp. 2–3

2.2. Eigenfunctions of Laplace-Beltrami operator

  • Laplace-Beltrami eigenfunctions are intrinsic to the geometry, and their spectral relationships connect them to geodesic distance and diffusion distance.Eigenfunctions form an orthogonal basis, with lower eigenvalues corresponding to lower-frequency functions.Source: §2.2 · p. 4
  • The method approximates eigenfunctions on a mesh by assembling finite-element Laplacian and mass matrices and solving Av = λMv.Source: §2.2 · p. 4

2.3. Physics-informed neural networks on manifolds

  • Each point is represented by the N eigenfunctions associated with the N lowest eigenvalues, with more eigenfunctions potentially needed for high-frequency problems.The eigenfunctions act as geometry-defined features with increasing frequency.Source: §2.3 · p. 5
  • Finite elements approximate derivatives of the network output; the resulting gradient is tangent to the manifold, and local operator evaluation supports mini-batch training.The approach applies operators to a piecewise-linear interpolant of network predictions at mesh nodes.Source: §2.3 · pp. 5–6
  • For PDEs, the method uses energy formulations, finite-element residuals, or a lumped mass matrix to obtain local operators.The choices depend on the problem; energy formulations are used for Poisson and hyperelasticity with conservative boundary conditions.Source: §2.3 · p. 6

3. Numerical experiments

The experiments test Δ-PINNs across six problems involving complex geometries and varied physics. Results show close agreement with reference solutions in several settings, while the Poisson study reveals sensitivity to eigenfunction count and discretization.

3.1. Eikonal equation on a coil

  • 3.68 · 10−3 normalized mean squared error for Δ-PINNs versus 0.98 for traditional PINNs and 2.95·10−2 for the graph convolutional network on the coil Eikonal problem.Δ-PINNs achieved high correlation on training and test data; removing the physics loss led to overfitting and poor test performance.Source: §3.1 · p. 7, Figure 3 · p. 7, Table 1 · p. 7

3.2. Heat transfer in a heat sink

  • Δ-PINNs most closely matched the heat-sink temperature distribution, distinguishing fins that are close in Euclidean space but far apart intrinsically.Traditional PINNs produced a nearly linear solution and failed to fit boundary data; signed-distance-function PINNs had artificial curvature at narrow fins.Source: Figure 4 · p. 8, §3.2 · p. 9

3.3. Poisson equation with exact eigenfunctions

  • 1.94 · 10−6 with exact eigenfunctions and 1.69 · 10−6 with numerical eigenfunctions were the minimum errors in the tested discretization range.The range was 0.066–0.1; results depended on eigenfunction count and discretization, while traditional PINNs achieved mean squared error 1.72 · 10−5. With 400 numerical eigenfunctions, element sizes above 0.1 were infeasible because there were too few nodes.Source: §3.3 · p. 10, Figure 7 · p. 11, Figure 6 · p. 11

3.4. ∆-PINNs on large meshes

  • 9.25 · 10−6 mean squared error was obtained on the large Lucy mesh, with 99,970 triangles and 49,987 nodes.Computing 100 eigenfunctions on the largest tested mesh, with 134,345 nodes, took less than 35 seconds; training remained the largest computational burden.Source: §3.4 · pp. 11–12, Figure 9 · p. 13

3.5. Hyperelasticity

  • 6.78 · 10−6 mean squared error was reported for the hyperelasticity problem, with excellent agreement to the reference solution.The example uses an ∩-shaped planar domain and demonstrates application to nonlinear mechanics.Source: §3.5 · pp. 13–14, Figure 10 · p. 14

3.6. Geodesic distance

  • 0.57% normalized mean absolute test error was achieved for geodesic distance on the bunny, with qualitatively good agreement in distances and paths.The test used 10,000 randomly selected pairs; displayed paths from 300 points to each of five origins were directed toward their origin points.Source: §3.6 · p. 15, Figure 11 · p. 15

4. Discussion

The discussion presents Laplace-Beltrami eigenfunctions as a geometry-aware encoding that improves PINN performance on complex-domain problems, especially when Euclidean and intrinsic distances diverge. It also identifies tuning and discretization constraints, while outlining extensions to 3D solids and time-dependent problems.

4. Discussion

  • ∆-PINNs outperform traditional PINNs especially when Euclidean-near points are far apart intrinsically, outperform a physics-informed graph network, and match a boundary-tailored SDF-PINN in reported comparisons.The paper tests inverse Eikonal, Poisson, heat-transfer and hyperelastic forward problems, plus geodesic-distance operator learning.Source: §4 · p. 16
  • Physics-informed eigenfunction inputs perform well, but eigenfunction-only regression overfits and depends heavily on eigenfunction count, consistent with physics providing regularization.The discussion also reports Gaussian processes as effective for manifold regression, while incorporating nonlinear physical constraints is cumbersome and retains their poor scaling.Source: §4 · p. 16
  • For Poisson problems, ∆-PINNs reach errors similar to traditional PINNs, and numerically computed eigenfunctions show no observed performance decrease versus exact ones.Accuracy varies with eigenfunction count; discretization size also affects accuracy non-monotonically, and its effects on training dynamics remain for future study.Source: §4 · p. 16
  • Finite-element operator approximation may require more sophisticated elements for operators beyond the gradient and Laplacian; eigenfunctions require preprocessing, which takes seconds on meshes over 10^5 nodes.The authors describe mesh-free eigenfunction and gradient approximations as possible future directions, not as part of the current approach.Source: §4 · p. 16
  • Future work proposes extending the method to 3D solid geometries and time-dependent problems by adding time as a neural-network feature.Source: §4 · p. 16

Appendix A. The use of finite elements in ∆-PINNs

The appendix derives a finite-element procedure for computing Laplace-Beltrami eigenfunctions and approximating PDE operators on a mesh. It describes the weak-form eigenproblem, its discretization and convergence, and the elementwise gradient construction used in the method.

Appendix A. The use of finite elements in ∆-PINNs

  • Finite elements approximate a function on a mesh using nodal shape functions and nodal values; the paper uses Lagrangian functions with a delta property and partition of unity.For surfaces, the linear case is piecewise linear over the mesh.Source: Appendix A · pp. 18–19
  • The Laplace-Beltrami eigenproblem is rewritten in variational form using test functions, with the boundary term vanishing under the stated boundary conditions.The described test-function spaces are H1(B) for Neumann conditions and H1_0(B) for Dirichlet conditions; the approximation uses piecewise-linear finite elements.Source: Appendix A · p. 19
  • The finite-element equations are assembled into a matrix system and solved as an eigenvalue problem to obtain discrete eigenfunctions.The appendix defines A as the assembly of local element matrices and represents the eigenfunction by its nodal values.Source: Appendix A · pp. 19–20
  • Under uniformity assumptions, discrete eigenfunctions converge to analytical ones as mesh size decreases; non-convexity and geometric approximation can limit accuracy.For linear elements on convex domains, eigenvalues converge quadratically with mesh size; high-curvature regions may incur geometric error.Source: Appendix A · p. 20
  • With linear triangular elements, the gradient is constant within each element and is computed from the element matrix B_e and nodal values.For surface gradients and the Laplace-Beltrami operator, the local-to-global rotation cancels; the Eikonal gradient norm likewise eliminates it because R_e^T R_e = I.Source: Appendix A · p. 20

Appendix B. Comparison to other methods

The appendix compares ∆-PINNs with a graph convolutional network and a boundary-condition-enforcing PINN based on signed distance functions.

Appendix B. Comparison to other methods

  • The comparison methods are a graph convolutional network [16] and a PINN that exactly imposes boundary conditions using signed distance functions [48].Source: Appendix B · p. 20

Appendix B.1. Graph convolutional neural network

The appendix contrasts local graph convolution with ∆-PINNs' encoding of global domain geometry, then compares their coil-problem performance. In the reported ablation, ∆-PINNs outperform the graph network and show less sensitivity to parameter count and initialization.

Appendix B.1. Graph convolutional neural network

  • Graph convolution propagates information locally across neighboring nodes, whereas ∆-PINNs encode global topological and geometric information using Laplace-Beltrami eigenfunctions.The passage notes that PDE solution operators are global even when discretized differential operators are sparse.Source: Appendix B.1 · pp. 20–21
  • With 1,546 coil-mesh nodes, ∆-PINNs train 9x faster than the graph convolutional network because graph outputs may depend on every node and prevent mini-batching.The authors expect the speed-up to grow on larger meshes, while ∆-PINN batch size can remain fixed.Source: Appendix B.1 · p. 21
  • Across coil ablations, ∆-PINNs outperform the graph convolutional network as data points increase, while the graph network's error rises at 120 points; ∆-PINNs are also less sensitive to parameter count and initialization.The data-count comparison uses fixed architectures; the parameter-count comparison trains with 60 points. Each run is repeated five times.Source: Figure B.12 · p. 21, Appendix B.1 · p. 22

Appendix B.2. Effect of the weight of partial differential equation loss in PINNs

For the coil Eikonal problem, varying the PDE-loss weight revealed an optimal value for traditional PINNs, but their error remained higher than ∆-PINNs. The network could still fit the data points when the PDE-loss weight was very small.

Effect of the weight of partial differential equation loss in PINNs

  • At w_r = 10^-2, traditional PINNs reached their best Eikonal result, yet their error remained an order of magnitude higher than ∆-PINNs.The comparison varied w_r from 10^-8 to 1 while keeping the data-loss weight at 1.Source: Appendix B.2 · p. 22, Figure B.13 · p. 22
  • With w_r = 10^-8, the data loss fell below 10^-5, showing that the network could fit the provided points.Source: Appendix B.2 · p. 22
  • Method comparisons kept parameter counts, training settings, and loss formulations as close as possible.Source: Appendix B.2 · p. 22

Appendix B.3. Exact imposition of boundary conditions with signed distance functions

The signed-distance-function approach imposes boundary conditions exactly, and improves on traditional PINNs in the heat-sink example, but does not match ∆-PINNs. Its demonstrated scope and handling of boundary data are limited.

Exact imposition of boundary conditions with signed distance functions

  • The construction combines a signed distance function that vanishes at the boundary with an interpolant for the boundary conditions, making them exact for any network parameters.Source: Appendix B.3 · p. 23
  • The signed-distance-function method significantly improves traditional PINNs in the heat-sink example but does not match ∆-PINNs.The comparison used three hidden layers of 100 neurons and the same number of training iterations as ∆-PINNs.Source: Appendix B.3 · p. 23
  • The approach has only been demonstrated on 2D examples, lacks a direct path to 3D surfaces, and exactly imposes noisy boundary measurements.The authors also note that extending it to 3D would be computationally expensive because each prediction requires iterating over all domain faces.Source: Appendix B.3 · p. 23

Appendix C. Regression with ∆-NN

The appendix tests ∆-NN regression on a dragon-shaped manifold without physics regularization and compares it with Gaussian process regression. ∆-NN accuracy depends on data availability and eigenfunction count, while Gaussian process regression performs better on this smooth target.

Regression with ∆-NN

  • The benchmark used a dragon mesh with 9,057 vertices and 18,110 triangles, and compared ∆-NN against a Matérn-kernel Gaussian process.The synthetic target was based on geodesic distance from a selected vertex.Source: Appendix C · p. 23
  • The overall mean squared error was 6.8 · 10^-2 for ∆-NN versus 1.8 · 10^-2 for Gaussian process regression on the 100-point dragon regression example.In areas without data, such as the horns, ∆-NN tended to predict more extreme values; where data were available, both methods approximated the function well.Source: Appendix C · p. 23, Figure C.14 · p. 24
  • Gaussian process regression with a Matérn kernel consistently had lower errors than ∆-NN when approximating the smooth geodesic-distance target.Source: Appendix C · p. 23
  • With few data points, fewer low-frequency eigenfunctions helped avoid overfitting; as data increased, more eigenfunctions reduced error, but 225 and 400 never performed well.The experiment used 50, 100, 500, or 1000 randomly selected data points and repeated sampling five times, reporting median error.Source: Appendix C · p. 23, Figure C.14 · p. 24

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