Enforcing Dirichlet Boundary Conditions in Operator Learning

Andrew M. Stuart; Margaret Trautner

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Enforcing Dirichlet Boundary Conditions in Operator Learning

Andrew M. Stuart, Margaret Trautner

arXiv:2608.27256v1Aug 27, 2026math.NAcs.LG

TL;DR

Neural operators commonly enforce known boundary conditions only approximately, while existing explicit approaches face grid and geometry restrictions. The paper uses Dirichlet Laplacian eigenfunctions in a kernel-integral architecture to enforce homogeneous conditions independently of training. It proves universal approximation and evaluates the method on Darcy flow and Helmholtz problems, with comparable training times and greater robustness to suboptimal hyperparameters reported in the experiments.

Problem
Common neural operator architectures learn boundary conditions from data only approximately, while explicit alternatives can require separable domains and Gaussian or uniform grids.
Method
The architecture constrains hidden-layer outputs to spans of homogeneous Dirichlet Laplacian eigenfunctions, enabling exact boundary enforcement without training in a general nonlocal neural operator.
Results
The method has universal approximation guarantees and is implemented on Darcy flow over a square and Helmholtz flow over a circular domain; training times were comparable between DNO and FNO in both examples.
Takeaways & Limitations
The approach supports arbitrary mesh data and general geometries while retaining the kernel-integral operator setting's approximation scope.
Takeaways & Limitations
Prior explicit boundary-enforcement methods may remain limited to separable domains or specific grid types, whereas the paper's broader theory assumes a bounded domain with Lipschitz boundary.

Abstract

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Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators thereby uniting existing theory for a variety of operator learning methods. We validate the proposed method on maps defined by the coefficient to solution map in 2D PDEs: Darcy flow on a square domain and the Helmholtz equation on a circular domain. Comparisons are made with alternative methods.

1. Introduction

Operator learning addresses infinite-dimensional maps arising in PDEs, but common architectures learn boundary conditions only approximately. The paper proposes explicit Dirichlet enforcement through Laplacian eigenfunctions while proving broad approximation results and evaluating the method on representative PDE operators.

Motivation: Operator learning approximates maps between infinite-dimensional function spaces, including PDE solution operators, from known equations or data.Neural operators include DeepONet, PCA-Net, random features, and FNO.
Motivation: Common neural operators rely on training data to enforce boundary conditions approximately rather than imposing known conditions explicitly.The paper identifies this limitation for architectures such as DeepONet and FNO.
Related work: Existing explicit methods use boundary-adapted bases but can require Gaussian or uniform grids, separable box-like domains, or restricted curved quadrilateral geometries.These restrictions limit applicability to more complex geometries and arbitrary mesh coordinate points.
Contributions: The paper proposes automatically satisfying Dirichlet boundary conditions independently of training by using Dirichlet Laplacian eigenfunctions within the operator architecture.The method is presented as a kernel-integral operator architecture with a Dirichlet eigenfunction layer.
Contributions: The authors prove universal approximation for a general class of nonlocal neural operators and implement the proposed method on Darcy flow and Helmholtz coefficient-to-solution maps.The experiments compare the method with FNO and other approaches that automatically satisfy boundary conditions.
Paper organization: The paper develops the architecture, theoretical results, numerical experiments, and supporting proofs across its main sections and appendices.The appendices contain technical neural-network lemmas, application-setting assumptions, main-lemma proofs, and Laplacian-eigenfunction results.

2. Definition of Nonlocal Neural Operator

The paper defines the nonlocal neural operator as a lifting–hidden-layer–projection architecture and specializes its basis functions to Laplacian Dirichlet eigenfunctions. This Dirichlet layer enables boundary enforcement while preserving the broader operator framework and its universal-approximation scope.

Nonlocal Neural Operator: The architecture uses basis functions in hidden layers, with common variants such as the Fourier Neural Operator included within this kernel-integral setting.The paper calls the NNO a kernel-integral neural operator and notes that FNO is a special case.
Nonlocal Neural Operator: The nonlocal neural operator maps input functions to output functions through learnable lifting, hidden, and projection layers.Its definition includes the lifting layer R, hidden layers L1, …, LL, and projection layer Q.
Basis Functions and Universality: The general universality theory permits basis functions satisfying mild conditions and allows finite subsets of suitable function families in the final hidden layer.The latter choice can impose desired output-function conditions while retaining universal approximation under the stated assumptions.
Dirichlet Layer: The Dirichlet layer sets its basis functions to Laplacian Dirichlet eigenfunctions on the domain, whose boundary behavior supports zero Dirichlet values.The construction is introduced as a modified NNO layer specifically designed for automatic boundary enforcement.
Dirichlet Layer: A final Dirichlet layer with zero bias and a zero-preserving projection layer enforces homogeneous Dirichlet conditions without training.Other hidden-layer choices may precede this final layer, and constant nonzero conditions can be handled through preprocessing and postprocessing.
Computational Complexity: A Dirichlet layer using M eigenfunctions has computational complexity O(NM), compared with O(N log N) for a hidden Fourier layer.Here N denotes the number of grid points, while M is the number of eigenfunctions used by the Dirichlet layer.

3. Theory

The theory establishes universality for a broad class of kernel-integral neural operators under general basis-function conditions, including architectures using Dirichlet eigenfunctions. It shows that the Dirichlet layer retains universal approximation while enforcing zero boundary conditions on bounded Lipschitz domains.

General universality: Theorem 3.4 proves universality for nonlocal neural operators using a single hidden layer and a single basis-function pair under the smoothed approximation assumption.The result applies to continuous operators on compact input sets and suitable basis functions.
General universality: Theorem 3.5 proves universality using basis functions already present in the NNO architecture, rather than reconstructing those basis functions with finite-dimensional neural networks.This theorem applies in the more general setting where approximation is available in C0(Ω) rather than a smoother function space.
Approximation assumptions: The smoothed approximation assumption requires a family of linear smoothers that converges uniformly, whose projections converge, and whose coefficient functionals are continuous.These conditions formalize approximation after smoothing the target output set.
Approximation assumptions: Proposition 3.2 verifies the assumption for several bases on bounded Lipschitz domains, including general smooth bases, Dirichlet Laplacian eigenfunctions, and tensorized Legendre polynomials.The Dirichlet-eigenfunction case covers compact subsets of C0(Ω).
Dirichlet layer: Corollary 3.6 shows that a final Dirichlet layer yields universal approximation for continuous operators into C0(Ω) while enforcing zero boundary conditions in the trace sense.The result assumes a bounded Lipschitz domain and uses Dirichlet Laplacian eigenfunctions as the final-layer basis.

4. Numerical Experiments

The experiments evaluate Dirichlet neural operators on Darcy flow over a square and Helmholtz problems over a circular domain, comparing them with FNO-based and regression alternatives. Results assess feasibility, cost-accuracy trade-offs, hyperparameter robustness, and data efficiency.

Experimental setup: The study tests absolute Dirichlet enforcement on coefficient-to-solution maps for Darcy flow and a Helmholtz problem.Darcy uses a square domain, while Helmholtz uses a circular domain.
Architecture and boundary treatment: The DNO uses a final Dirichlet layer with W_l = 0 and b_l = 0 to enforce zero boundary values.For Helmholtz, the output is transformed to u − b1 before learning and shifted back during postprocessing.
Baselines: The comparisons include standard FNO, FNO projection before or during training, and ridge regression on projected input-output coefficients.The ridge-regression baseline is deterministic, whereas the neural models are trained stochastically.
Data generation: The Darcy dataset contains 2048 samples generated on a 421 × 421 grid, with 248 validation and 200 held-out test samples.The coefficients are generated from a transformed Gaussian measure, and solutions use a second-order finite-difference scheme.
Data generation: The Helmholtz data use finite-element solutions on a triangular unit-disk mesh, which the DNO receives directly while FNO data are interpolated to a masked uniform grid.The DNO therefore accommodates the original triangular mesh, whereas FNO requires a box-domain representation.
Hyperparameter search: The Darcy search selected 12 modes, width 64, 300 epochs, batch size 15, four FNO layers, and one final DNO layer; FNO and DNO training times were comparable.Figures 3 and 4 visualize relative validation error against training time for the two searches.
Results and discussion: The DNO was more robust to suboptimal hyperparameters and improved Darcy accuracy by 2× in the low-data regime and 20% over FNO at 1600 training points.Other nonlinear choices failed to beat ridge regression in the low-data regime, while DNO retained an advantage with more data.

5. Conclusion

The paper introduces a Laplacian-eigenfunction basis that enforces Dirichlet conditions automatically and supports arbitrary meshes and domains under Lipschitz-boundary assumptions.

The proposed basis uses Laplacian eigenfunctions so the nonlocal neural operator automatically satisfies Dirichlet boundary conditions.
The approximation theory extends beyond this setting to a broad class of L2 bases on domains with Lipschitz boundaries.
The authors implement the Dirichlet Neural Operator on Darcy flow over a square and Helmholtz flow over a circular domain.
The method enforces boundary conditions without training on arbitrary meshes and domains, distinguishing it from prior work.
The work identifies automatic enforcement of mixed boundary conditions on arbitrary domains as an open question.

Appendix A. Properties of finite-dimensional neural networks

This appendix defines finite-dimensional feedforward neural networks and states their standard approximation capability for smooth, non-polynomial activations.

A finite-dimensional feedforward neural network is defined as a composition of affine maps and a component-wise activation function.
The activation is assumed to be C^s and non-polynomial for some nonnegative integer s.
For a bounded domain and a C^s target function, a neural network with non-polynomial C^s activation can approximate the target within any ϵ > 0.

Appendix B. Proof of Proposition 3.2

The proof establishes approximation properties for smoothed functions and their projections onto Laplacian, polynomial, and Fourier-type bases under the stated regularity and convergence conditions.

Periodic extension, mollification, and Fourier-series approximation provide the corresponding convergence argument in the periodic setting.
The proof uses smoothing and cutoff constructions to approximate functions while enforcing zero values near the boundary.
Equicontinuity and compactness control the approximation uniformly over the relevant function family.
For Laplacian eigenfunctions, continuity and projection convergence establish the required approximation property.
Tensorized Legendre polynomials form an orthonormal basis on the reference domain, with differential-operator estimates used in the proof.
The estimates require choosing an integer k sufficiently large for the relevant series to converge.

Appendix C. Proof of Lemma 3.8

The proof constructs continuous functionals after smoothing inputs and approximates the target operator uniformly over compact sets using the assumed basis approximation properties.

Lemma 3.8 assumes a continuous operator on a bounded Lipschitz domain whose image satisfies the Smoothed Approximation Assumption.
The constructed approximation uses finitely many continuous functionals from L1 inputs to represent the operator output.
Choosing the smoothing scale sufficiently small controls the approximation error before selecting a finite basis truncation.
Input smoothing maps L1 functions continuously into C^s functions and converges uniformly on the compact input set.
The conclusion combines the smoothing and finite-dimensional approximation steps to obtain the desired operator approximation.

Appendix D. Proofs of Lemma 3.10 and 3.11

The appendix introduces a lemma showing that certain integrals can be approximated by finite-dimensional neural networks weighted by a nonnegative function. The construction assumes bounded inputs and controlled continuous weighting data.

Lemma D.1: Lemma D.1 approximates certain integrals using a finite-dimensional neural network weighted by a positive function.The weighting function is continuous, bounded, nonnegative, and vanishes only on a measure-zero set.
Lemma D.1: The lemma considers a bounded domain, a compact set of L1-valued inputs with uniformly bounded L∞ norms, and fixed continuous L1 data.These assumptions provide the boundedness and regularity conditions needed for approximation.
Lemma D.1: The weighting function may be normalized without loss of generality because the final network layer can absorb the rescaling.Its nonnegativity and almost-everywhere positivity are used in the normalization argument.

4. Then define

The proofs establish approximation results for nonlocal neural operators by mollifying inputs, projecting onto finite-dimensional bases, and approximating the resulting continuous finite-dimensional maps with neural networks. A cutoff and exact positive channel control boundary behavior and yield the target error bound.

Network construction: An exactly realized positive channel enables exact division by the basis function, while ordinary neural-network universality approximates the remaining continuous maps.The denominator is bounded below by η− > 0, supporting the Cs′ error estimate.
Lemma statements: Lemmas 3.10 and 3.11 approximate continuous functionals on compact bounded subsets of L1 by nonlocal neural operators with Cs′ outputs.The domain is bounded and Lipschitz, while the target regularity is indexed by the integer s′.
Shared proof: The proof first extends inputs by zero, mollifies them, and uses compactness to obtain uniformly accurate regularized functionals.Continuity of α gives uniform convergence over the compact input set as the mollification scale decreases.
Shared proof: The regularized inputs are projected onto finitely many smooth basis functions, reducing the functional approximation to finitely many inner products.The truncation dimension J is chosen sufficiently large after selecting the mollification parameter δ.
Network construction: The network approximates the coefficient-dependent integrals by parallel finite-dimensional subnetworks and lifts their outputs together with constant and target-basis channels.The lifting layer explicitly includes the coefficient approximations, a constant 1, and Rη(x).
Error bound: The approximation errors are allocated across the construction so that their sum is bounded by the prescribed ε.The proof selects δ, J, and subsequent network tolerances sequentially to satisfy the final estimate.

Appendix E. Eigenfunctions of the Laplacian

Appendix E proves uniform convergence of projections onto homogeneous-Dirichlet Laplacian eigenfunctions for sufficiently regular compactly supported functions. The proof controls projection tails through repeated Laplacian integration by parts, eigenfunction bounds, and Weyl’s law.

Assumption: Compact support is essential because Laplacian-eigenfunction projection nullifies the projected function and every iterated Laplacian at the boundary.This boundary behavior is part of why the proof assumes the approximated set lies in Ck_c(Ω).
Lemma E.1: Lemma E.1 considers bounded sets of compactly supported Ck functions on bounded Lipschitz domains, with k > d + 1.The projection basis consists of Laplacian eigenfunctions satisfying homogeneous Dirichlet boundary conditions.
Projection result: The projection PJw = Σj=1^J⟨w, ηj⟩L2(Ω)ηj onto the first J eigenfunctions converges uniformly in C(Ω) as J →∞.The eigenfunctions form the approximation basis used in the convergence result.
Proof strategy: The proof bounds projection tails using repeated Laplacians, eigenfunction estimates, and Weyl’s law for counting eigenvalues.The argument handles repeated eigenvalues by grouping indices associated with the same eigenvalue.