cs.LGSep 28, 2026

KernelOnet: An Interpretable Neural Operator Based on Kernel Functions

Authors: Yuan Guo, Hanshu Chen, Qiang Xi, Timon Rabczuk, Zhuojia Fu

Organizations: College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China · Institute of Structural Mechanics, Bauhaus-University Weimar, Weimar 99423, Germany · Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing 210098, China

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Abstract

This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitly into the neural operator architecture, so that the operator structure matches the kernel-expansion form used in boundary-type kernel-expansion methods. Unlike traditional neural operators such as DeepONet, which learn basis functions implicitly through deep networks, KernelOnet replaces the trunk network with explicit kernels and offers three complementary kernels: a data-driven learnable kernel, in which a neural network parameterizes a radial basis function learned from data, and which for constant-coefficient linear problems can be regarded as a non-singular fundamental solution; a physics-informed kernel, which embeds physical information such as analytic fundamental solutions into the network structure, so that the expansion satisfies the governing equation automatically and can be trained without supervision on boundary conditions alone, with no interior solution data; and a hybrid kernel, which splits the solution, according to the linear principal part of the governing equation, into a homogeneous part spanned by analytic fundamental solutions and a source part carried by low-rank learned correction kernels, thereby balancing physical priors against data fitting on nonlinear problems lacking an analytic fundamental solution. On three benchmarks and one engineering problem in a shallow-water waveguide, KernelOnet attains high accuracy; where comparable with DeepONet, it is more accurate with fewer learnable parameters. Its unsupervised configuration needs no interior solution labels, and its per-query inference cost is far below that of per-instance solvers, offering an effective route to acoustic propagation in unbounded exterior domains that general-purpose neural operators struggle to handle.

Figures & tables

Figure 1: Architectures of DeepONet and PI-DeepONet. DeepONet learns the basis functions tkt_{k} implicitly through a fully connected trunk network and optimizes the data loss in a supervised manner; on this basis, PI-DeepONet adds the PDE residual and the boundary condition residual to the loss function as soft constraints, so that in the unsupervised case the loss contains only these two terms, balanced by the weights λPDE\lambda_{\mathrm{PDE}} and λBC\lambda_{\mathrm{BC}} , while the data loss term can be added when supervised data are available to further improve performance.
Figure 2: Architecture of KernelOnet. The three variants share the same branch network and the same operator expansion structure u(x)=∑j=1Bbj(a) ψj(x)u(\mathbf{x})=\sum_{j=1}^{B}b_{j}(a)\,\psi_{j}(\mathbf{x}) , differing only in the kernel trunk ψj(x)\psi_{j}(\mathbf{x}) : KernelOnet-RBF uses a neural-network-parameterized radial basis function; KernelOnet-PIKF takes the analytic fundamental solution as its kernel (the collocation form uses a γ\gamma -shifted fundamental solution, the boundary integral form a single-layer potential kernel on the true boundary) and its expansion satisfies the governing equation exactly, so that it can be trained without supervision; KernelOnet-HK mixes analytic fundamental-solution kernels with low-rank learned correction kernels, ψ=Φ(γ xb)⊕φθ(x−tk)\psi=\Phi(\gamma\,\mathbf{x}_{b})\oplus\varphi_{\theta}(\mathbf{x}-\mathbf{t}_{k}) , absorbing the nonlinear source terms into a correction branch with Kc≪BK_{c}\ll B and requiring supervised training.
 DeepONetPI-DeepONetKernelOnet-RBFKernelOnet-PIKFKernelOnet-HK
Form of basis functionsImplicitly learned basis functionsImplicitly learned basis functionsNN-parameterized radial basis function ψ(r)=φθ(r)\psi(r)=\varphi_{\theta}(r)Analytic fundamental-solution kernel: collocation / boundary integralAnalytic fundamental solution ++ low-rank correction kernel Φ⊕φθ\Phi\oplus\varphi_{\theta}
Embedding of physical informationNoneSoft-constrained PDE residualData-driven kernel functionHard-coded into the kernel functionHard-coded ++ low-rank source-term correction
PDE consistencyNot satisfiedApproximately satisfied, collocation onlyNot satisfied in advanceExactly satisfied, Lu=0\mathcal{L}u=0 in the domainApproximately satisfied, homogeneous branch exact and correction branch carrying the source term
Supervision informationInterior solution labelsBoundary conditions ++ PDE residualInterior solution labelsBoundary conditions only, unsupervisedInterior solution labels
Learnable kernel parametersNoneNoneShallow kernel function networkCollocation: 1 scalar γ\gamma ; integral: noneLow-rank correction network φθ\varphi_{\theta}
Nonlinear problemsApplicableApplicableApplicableNot applicableApplicable
Table 1: Comparison of DeepONet, PI-DeepONet, KernelOnet-RBF, KernelOnet-PIKF and KernelOnet-HK.
 KernelOnet-PIKFKernelOnet-HKKernelOnet-RBF
Kernel spaceAn approximation subspace of the homogeneous solution space (spanned by fundamental solutions)VB⊕VKc\mathcal{V}_{B}\oplus\mathcal{V}_{K_{c}} : fundamental-solution subspace ++ low-rank correction subspaceNeural-network-parameterized radial basis function
Approximation errorCan be made arbitrarily small; the space is dense in the solution spaceHomogeneous branch minimal (spectral convergence) ++ source branch decaying with KcK_{c} according to compressibilityCan be large; depends on the learned kernel shape
Estimation errorBoundary fitting residual onlyBoundary fitting of the homogeneous branch ++ interior/residual fitting of the correction branchData fitting ++ kernel shape learning
ConvergenceSpectral or nearly exponentialSpectral for the homogeneous branch; total error improves with KcK_{c} and the compressibility of the source termSpectral or algebraic, depending on kernel smoothness
Table 2: Comparison of KernelOnet-RBF, KernelOnet-PIKF and KernelOnet-HK from the perspective of kernel function theory.
MethodLearnable parametersRelative L2L_{2}Time per 100 epochs (s)
DeepONet180,8011.89×10−31.89\times 10^{-3}0.08
PI-DeepONet180,8013.53×10−33.53\times 10^{-3}0.53
KernelOnet-RBF129,2811.29×10−31.29\times 10^{-3}0.43
KernelOnet-PIKF103,0418.89×10−48.89\times 10^{-4}0.06
KernelOnet-HK132,5032.04×10−32.04\times 10^{-3}0.10
Table 3: Case 1: comparison of the training results of DeepONet, PI-DeepONet, KernelOnet-RBF, KernelOnet-PIKF, and KernelOnet-HK.
Figure 3: Case 1: predicted solutions and absolute error fields for a representative test sample. (a) MFS reference solution. (b) Distribution of the boundary points (red crosses, used to encode the Dirichlet data and generate the kernel centers) and the interior evaluation points (blue dots). (c)–(g) Predicted solutions and absolute errors of DeepONet, KernelOnet-PIKF, PI-DeepONet, KernelOnet-HK, and KernelOnet-RBF, with the corresponding relative L2L_{2} errors annotated. Note that KernelOnet-PIKF and PI-DeepONet did not use any interior solution labels during training.
Figure 4: Case 1: convergence history of the training loss. The left panel shows the supervised methods (DeepONet, KernelOnet-RBF, KernelOnet-HK), whose loss is the mean squared error relative to the interior reference solution; the right panel shows the unsupervised methods (KernelOnet-PIKF and PI-DeepONet), whose losses are the boundary residual and the weighted sum of the PDE residual and the boundary residual, respectively.
Figure 5: Case 1: kernel diagnostics and functional verification of the learned kernel. (a) The data-driven kernel ψ(r)=φθ(r)\psi(r)=\varphi_{\theta}(r) of KernelOnet-RBF (red), which is well described by the affine fit 1.05 Φ(r)−0.401.05\,\Phi(r)-0.40 (green dashed line), i.e., the network recovers the fundamental solution Φ(r)=−12πln⁡r\Phi(r)=-\frac{1}{2\pi}\ln r up to a scale and an offset, while remaining finite at the origin with ψ(0)=0.477\psi(0)=0.477 . (b) The NN-based MFS obtained by taking this learned kernel as the fundamental solution and collocating on the boundary of the inscribed square: left, the analytical solution u∗=x3−3xy2u^{*}=x^{3}-3xy^{2} ; middle, the predicted solution; right, the absolute error field (the relative L2L_{2} error is given in the title); the source points are 160160 boundary points on the circle of radius 0.50.5 , and the solution domain is the square inscribed in this circle.
Figure 6: Case 2: relative L2L_{2} error as a function of the nonlinearity strength ε\varepsilon . HK with the correction branch switched off ( Kc=0K_{c}=0 ) deteriorates rapidly with ε\varepsilon , followed by KernelOnet-RBF, whereas HK with the low-rank correction branch retained hardly degrades and surpasses RBF for ε≥1\varepsilon\geq 1 .
Figure 7: Case 2: (a) comparison of the kernel ψ(r)=φθ(r)\psi(r)=\varphi_{\theta}(r) learned by KernelOnet-RBF (red) with the analytic fundamental solution Φ(r)=12πK0(kr)\Phi(r)=\frac{1}{2\pi}K_{0}(kr) (blue) at ε=0.1\varepsilon=0.1 , where the red dot is ψ(0)\psi(0) ; (b) the learned kernels φθ(r)\varphi_{\theta}(r) at ε=1,2,3,4\varepsilon=1,2,3,4 , whose shapes are similar while their amplitudes increase with the nonlinearity strength.
Figure 8: Case 2: (a) FEM reference solution; (b) finite element mesh ( 40×20040\times 200 , central fan triangulation); (c)–(e) predicted solutions (left) and absolute error fields (right) of DeepONet, KernelOnet-RBF, and KernelOnet-HK, where the titles of the error plots give the relative L2L_{2} error for this sample.
Wavenumber kk55101015152020
KernelOnet-PIKF (collocation)1.14×10−31.14\times 10^{-3}1.37×10−31.37\times 10^{-3}2.64×10−32.64\times 10^{-3}1.90×10−31.90\times 10^{-3}
KernelOnet-PIKF-SVD (collocation ++ SVD)1.12×10−31.12\times 10^{-3}1.25×10−31.25\times 10^{-3}1.30×10−31.30\times 10^{-3}1.36×10−31.36\times 10^{-3}
KernelOnet-PIKF-SL (boundary integral)2.62×10−32.62\times 10^{-3}2.49×10−32.49\times 10^{-3}3.25×10−33.25\times 10^{-3}5.15×10−35.15\times 10^{-3}
KernelOnet-PIKF-SL-SVD (boundary integral ++ SVD)1.31×10−31.31\times 10^{-3}1.40×10−31.40\times 10^{-3}1.56×10−31.56\times 10^{-3}1.85×10−31.85\times 10^{-3}
Table 4: Case 3: average relative L2L_{2} error (averaged over the real/imaginary parts) of the two kernel-basis forms, collocation and boundary integral, and of their SVD-truncated variants, on the complex Helmholtz problem in an unbounded exterior domain, as a function of the wavenumber kk . The SVD truncation order is selected offline according to the boundary reconstructability criterion: q=40q=40 for the collocation form at all wavenumbers, and q=40,40,75,110q=40,40,75,110 for the boundary integral form at k=5,10,15,20k=5,10,15,20 , respectively.
Figure 9: Case 3: predicted solution and absolute error field for a representative test sample at k=20k=20 (KernelOnet-PIKF, collocation form). The top and bottom rows show the real and imaginary parts of the solution, respectively, and the three columns give in turn the MFS reference solution, the KernelOnet-PIKF predicted solution, and the absolute error field annotated with the relative L2L_{2} error.
Figure 10: Case 4: schematic of the numerical model for shell structural acoustic radiation in a shallow-water waveguide (not to scale). The origin of the rectangular coordinates (x,y,z)(x,y,z) lies at the sea surface, with the zz axis pointing vertically upward; Ωo\Omega_{o} is the seawater domain (density ρ1\rho_{1} , sound speed c1c_{1} ); the sea surface is a pressure-release boundary Γ1\Gamma_{1} , and at a depth HH below it lies a penetrable boundary Γ2\Gamma_{2} , beneath which is the seafloor sediment layer (density ρ2\rho_{2} , sound speed c2c_{2} ). This case takes a spherical-shell radius R=0.5R=0.5 m, sphere-center depth h=15h=15 m, sea depth H=25H=25 m, and frequency f=240f=240 Hz; the sea-surface reflection coefficient is a2=−1a_{2}=-1 and the seafloor reflection coefficient is a1=0.4626a_{1}=0.4626 .
Figure 11: Case 4 (near-field acoustic radiation in a shallow-water waveguide, x∈[2,50]x\in[2,50] m, z∈[−24,−1]z\in[-24,-1] m): reference solution (left), unsupervised KernelOnet-PIKF prediction (middle), and relative amplitude error (right) for a representative test sample. The left and middle columns show the sound pressure level SPL =20log⁡10(∣p∣/10−6)=20\log_{10}(|p|/10^{-6}) (dB), and the right column shows the relative amplitude error ε~=∣∣ppred∣−∣pref∣∣/∣pref∣\tilde{\varepsilon}=\bigl||p_{\mathrm{pred}}|-|p_{\mathrm{ref}}|\bigr|/|p_{\mathrm{ref}}| . The shallow-water waveguide interference structure formed by multiple reflections at the sea surface and the seafloor is clearly visible, and the prediction is visually almost indistinguishable from the reference solution.
Solution configurationKernel functionRelative L2L_{2}Relative amplitude errorPhase error ( ∘ )Transmission loss deviation (dB)
Near field KernelOnet-PIKFWaveguide Pekeris GnG^{n} (ring source)8.57×10−48.57\times 10^{-4}7.17×10−47.17\times 10^{-4}3.2×10−23.2\times 10^{-2}6.7×10−36.7\times 10^{-3}
Far field KernelOnet-PIKF ( c1(z)=1507−0.24zc_{1}(z)=1507-0.24z )Normal mode GfG^{f}1.18×10−31.18\times 10^{-3}9.04×10−49.04\times 10^{-4}3.9×10−23.9\times 10^{-2}9.1×10−39.1\times 10^{-3}
Far field KernelOnet-PIKF ( c1(z)=1510c_{1}(z)=1510 )Normal mode GfG^{f}1.31×10−31.31\times 10^{-3}1.02×10−31.02\times 10^{-3}4.1×10−24.1\times 10^{-2}7.5×10−37.5\times 10^{-3}
Far field KernelOnet-PIKF ( c1(z)=1513+0.24zc_{1}(z)=1513+0.24z )Normal mode GfG^{f}1.18×10−31.18\times 10^{-3}8.80×10−48.80\times 10^{-4}3.9×10−23.9\times 10^{-2}4.4×10−34.4\times 10^{-3}
Table 5: Case 4: training results with the unsupervised KernelOnet-PIKF for both the near field and the far field under the various sound speed profiles. The near field uses the collocation form with truncation order q=40q=40 , and the far field selects the retained directions of the kernel basis according to the input subspace; the relative L2L_{2} error is reported separately for the real and imaginary parts and then averaged, and the definitions and statistical conventions of the relative amplitude error, the phase error, and the transmission loss deviation are given in the main text.
Figure 12: Case 4 (far-field acoustic propagation using the normal-mode kernel, x∈[950,1000]x\in[950,1000] m, z∈[−24,−1]z\in[-24,-1] m): reference solution (left column), unsupervised KernelOnet-PIKF prediction (middle column), and relative amplitude error (right column) for a representative test sample, corresponding to three shallow-water sound speed profiles c1(z)c_{1}(z) . Here zz is the vertical coordinate (sea surface z=0z=0 , seafloor z=−Hz=-H ); the left and middle columns show the sound pressure level SPL (dB), and the right column shows the relative amplitude error ε~=∣∣ppred∣−∣pref∣∣/∣pref∣\tilde{\varepsilon}=\bigl||p_{\mathrm{pred}}|-|p_{\mathrm{ref}}|\bigr|/|p_{\mathrm{ref}}| . The three profiles: (a) c1(z)=1507−0.24zc_{1}(z)=1507-0.24z , (b) c1(z)=1510c_{1}(z)=1510 , and (c) c1(z)=1513+0.24zc_{1}(z)=1513+0.24z . Under all three profiles, the operator accurately reproduces the far-field waveguide modal interference structure (nodes and antinodes along the depth direction); the figure shows a single test sample, and the average errors over the test set are given in Table 5 .
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Operator L\mathcal{L}2D3D
Δ\Delta (Laplace)−12πln⁡r-\dfrac{1}{2\pi}\ln r14πr\dfrac{1}{4\pi r}
Δ+k2\Delta+k^{2} (Helmholtz)i4H0(1)(kr)\dfrac{i}{4}H_{0}^{(1)}(kr)e−ikr4πr\dfrac{e^{-ikr}}{4\pi r}
Δ−k2\Delta-k^{2} (modified Helmholtz)12πK0(kr)\dfrac{1}{2\pi}K_{0}(kr)e−kr4πr\dfrac{e^{-kr}}{4\pi r}
DΔ+v⋅∇−k2D\Delta+\mathbf{v}\cdot\nabla-k^{2} (convection–diffusion)12πK0(μr) e−v⋅r2D\dfrac{1}{2\pi}K_{0}(\mu r)\,e^{-\frac{\mathbf{v}\cdot\mathbf{r}}{2D}}e−μr−v⋅r2D4πr\dfrac{e^{-\mu r-\frac{\mathbf{v}\cdot\mathbf{r}}{2D}}}{4\pi r}
Δ2\Delta^{2} (biharmonic)−r2ln⁡r8π-\dfrac{r^{2}\ln r}{8\pi}r8π\dfrac{r}{8\pi}
Appendix
Table A1: Fundamental solutions Φ\Phi of common differential operators (2D and 3D).
CaseMethodrel. L2L_{2}rel. H1to 5×10−25\times 10^{-2}to 10−210^{-2}to 5×10−35\times 10^{-3}to 10−310^{-3}
1DeepONet1.89×10−31.89\times 10^{-3}3.47×10−23.47\times 10^{-2}9,500105,000182,500–
1PI-DeepONet3.53×10−33.53\times 10^{-3}4.17×10−24.17\times 10^{-2}21,500225,000389,000–
1KernelOnet-RBF1.29×10−31.29\times 10^{-3}3.10×10−23.10\times 10^{-2}3,00014,50040,000–
1KernelOnet-PIKF8.89×10−48.89\times 10^{-4}6.30×10−36.30\times 10^{-3}1,50010,50020,000379,500
1KernelOnet-HK2.04×10−32.04\times 10^{-3}2.19×10−22.19\times 10^{-2}3,00029,00093,000–
2 ( ε=4\varepsilon=4 )DeepONet5.71×10−35.71\times 10^{-3}7.06×10−27.06\times 10^{-2}19,500275,000––
Appendix
Table D1: Relative L2L_{2} and relative H1H^{1} semi-norm errors of each case and each method, together with the numbers of training epochs required for the validation error to first fall to given accuracy levels, all computed over 20002000 test samples; “ – ” means that the level was not reached within the 5×1055\times 10^{5} -epoch budget.Case 2 takes ε=4\varepsilon=4 and also lists the Kc=0K_{c}=0 baseline with the correction branch disabled; Case 3 takes the main case k=20k=20 ; for Case 4 the three sound speed profiles are listed for the far field.
CaseMethodLearnable parameterss/100 ep.Training per sample ( μ\mu s)Memory, training (MiB)Memory, inference (MiB)
1DeepONet180,8010.0760.3812821
1PI-DeepONet180,8010.5342.6768721
1KernelOnet-RBF129,2810.4312.15689332
1KernelOnet-PIKF103,0410.0610.304820
1KernelOnet-HK132,5030.1020.5116959
2 ( ε=4\varepsilon=4 )DeepONet187,2010.0740.3710320
Appendix
Table D2: Learnable parameters, training cost, and GPU memory usage of each case and each method. The per-sample training time is obtained by dividing the time per 100100 epochs by 100×2000100\times 2000 . Case 2 takes the nonlinear main case ε=4\varepsilon=4 and also lists the Kc=0K_{c}=0 baseline with the correction branch disabled; Case 3 takes the main case k=20k=20 ; both the near field and the far field of Case 4 are trained unsupervised, and the three sound speed profiles are listed for the far field.
CaseMethodRef. soln. generation (s/2000)Per-instance solve (ms)Inference (ms/sample)Training wall clock (h)N⋆N^{\star}
1DeepONet0.150.150.032/1.020.032/1.020.0860.0860.1060.106−−/4.1×105--/4.1\times 10^{5}
1PI-DeepONet †0.150.150.032/1.020.032/1.020.0850.0850.7410.741−−/2.9×106--/2.9\times 10^{6}
1KernelOnet-RBF0.150.150.032/1.020.032/1.021.5151.5150.5990.599−−/−−--/--
1KernelOnet-PIKF †0.150.150.032/1.020.032/1.020.0670.0670.0840.084−−/3.2×105--/3.2\times 10^{5}
1KernelOnet-HK0.150.150.032/1.020.032/1.020.1830.1830.1420.142−−/6.1×105--/6.1\times 10^{5}
2 ( ε=4\varepsilon=4 )DeepONet24452445102210220.0830.0830.1030.1033.6×1023.6\times 10^{2}
Appendix
Table D3: Reference-solution generation cost, the inference cost of the classical per-instance solver and of the neural operator, and the break-even number of queries N⋆N^{\star} for each case. All times are measured on the machine used in this paper, averaged over 100100 instances. Solvers: the method of fundamental solutions (MFS) for Cases 1 and 3, finite elements ++ Newton for Case 2, a ring-source boundary integral for the near field and normal modes for the far field of Case 4. The per-instance solve and N⋆N^{\star} are both given in the pre-factorized and the from-scratch settings; Case 2 is a nonlinear problem whose Jacobian changes with the solution and which has no pre-factorized setting, so only a single value is listed. “ – ” means that the single-instance cost of the classical solver is below the operator inference cost, in which case no break-even point exists; † denotes unsupervised training that uses no interior solution labels, whose training does not depend on the reference-solution generation cost.
MethodBatch sizerel. L2L_{2}rel. H1s/100 ep.Total wall clock (h)Peak memory (MiB)
DeepONet5002.15×10−32.15\times 10^{-3}2.18×10−22.18\times 10^{-2}0.2600.2600.3620.3626161
DeepONet10002.82×10−32.82\times 10^{-3}3.47×10−23.47\times 10^{-2}0.1360.1360.1880.1887373
DeepONet15003.03×10−33.03\times 10^{-3}3.63×10−23.63\times 10^{-2}0.1370.1370.1900.1908989
DeepONetFull batch (2000)5.71×10−35.71\times 10^{-3}7.06×10−27.06\times 10^{-2}0.0740.0740.1030.103103103
KernelOnet-RBF5003.03×10−33.03\times 10^{-3}2.98×10−22.98\times 10^{-2}1.5871.5872.2042.204630630
KernelOnet-RBF10003.81×10−33.81\times 10^{-3}3.87×10−23.87\times 10^{-2}0.8120.8121.1271.127633633
Appendix
Table D4: Batch-size ablation for the three data-driven models on Case 2 ( ε=4\varepsilon=4 ): relative L2L_{2} /H1 semi-norm errors, training time per 100100 epochs, total training wall-clock time, and peak training GPU memory. The full batch means that the batch size equals the training-set size, and its values agree with the corresponding rows of this case in Table D2 .
Evaluation grid nr×nθn_{r}\times n_{\theta}10×1010\times 1020×2020\times 2040×4040\times 40 (training)80×8080\times 80160×160160\times 160
DeepONet2.35×10−32.35\times 10^{-3}2.07×10−32.07\times 10^{-3}1.89×10−31.89\times 10^{-3}1.80×10−31.80\times 10^{-3}1.76×10−31.76\times 10^{-3}
KernelOnet-RBF1.50×10−31.50\times 10^{-3}1.35×10−31.35\times 10^{-3}1.29×10−31.29\times 10^{-3}4.36×10−34.36\times 10^{-3}4.31×10−34.31\times 10^{-3}
KernelOnet-HK2.50×10−32.50\times 10^{-3}2.19×10−32.19\times 10^{-3}2.04×10−32.04\times 10^{-3}1.96×10−31.96\times 10^{-3}1.92×10−31.92\times 10^{-3}
Appendix
Table D5: Resolution invariance of the three data-driven models with respect to the evaluation grid on Case 1: relative L2L_{2} error as a function of the evaluation grid nr×nθn_{r}\times n_{\theta} . All models are trained on the native 40×4040\times 40 grid and then evaluated directly on each grid with frozen weights; the reference solution is sampled from the same fundamental solution on each grid, and the average over 20002000 test samples is reported. The number of evaluation-grid points increases from 100100 to 25,60025{,}600 , a span of 256256 times.
WindowReference point (r0,z0)(r_{0},z_{0}) (m)TL at source depth (dB)TL in window (dB)Reference point relative to peak (dB)
Near field ( x∈[2,50]x\in[2,50] m)(2.0, 14.7)(2.0,\,14.7)22 m: 00 ; 26.826.8 m: +23.0+23.0 ; 5050 m: +21.2+21.2−18.0-18.0 / +20.8+20.8 / +81.7+81.7−1.1-1.1
Far field ( c1(z)=1507−0.24zc_{1}(z)=1507-0.24z )(950.0, 14.6)(950.0,\,14.6)950950 m: 00 ; 975975 m: −1.8-1.8 ; 10001000 m: −2.5-2.5−53.0-53.0 / −11.3-11.3 / +43.7+43.7−13.0-13.0
Far field ( c1(z)=1510c_{1}(z)=1510 )(950.0, 14.6)(950.0,\,14.6)950950 m: 00 ; 975975 m: −1.6-1.6 ; 10001000 m: −1.6-1.6−37.4-37.4 / −9.6-9.6 / +51.1+51.1−12.7-12.7
Far field ( c1(z)=1513+0.24zc_{1}(z)=1513+0.24z )(950.0, 14.6)(950.0,\,14.6)950950 m: 00 ; 975975 m: −0.1-0.1 ; 10001000 m: +0.5+0.5−39.7-39.7 / −7.4-7.4 / +52.0+52.0−8.8-8.8
Appendix
Table D6: Transmission loss of the reference solution of Case 4 in each evaluation window, over 20002000 test samples. TL=−20log⁡10(∣p(r,z)∣/∣p(r0,z0)∣)\mathrm{TL}=-20\log_{10}\bigl(|p(r,z)|/|p(r_{0},z_{0})|\bigr) , where the reference point (r0,z0)(r_{0},z_{0}) is taken at the source depth at the nearest distance within the window; the TL at the source depth is the sample mean, and the TL within the window is listed as the minimum, median, and maximum over the pooled set of samples and window points.
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Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks

Operator learning methods such as DeepONets and FNOs often struggle with PDE families featuring sharp interfaces, heterogeneous coefficients, and localized multiscale structures. We introduce a partition-of-unity (POU) mixture-of-experts framework for localized operator learning, in which geometry-aware gating networks produce smooth spatial partitions which blend the contributions of local expert networks. Our main contribution is HiRefPOU, a residual-style hierarchical POU architecture for DeepONets that organizes localized representations through nested parent-child partitions while preserving global continuity. We also show that the same POU principle can be incorporated into Fourier Neural Operators to introduce spatial adaptivity without modifying the underlying spectral layers. On heterogeneous Darcy and reaction-diffusion benchmarks, HiRefPOU achieves substantially lower error than global DeepONet and static POU-MoE baselines, while the broader operator-learning experiments show that the benefits of localization depend on the PDE structure and the chosen neural-operator backbone. The learned partitions are interpretable and align with interfaces and regions of rapid solution variation. These results show that explicit geometric localization can improve both accuracy and interpretability in neural operator learning.

Madison Cooley, Ramansh Sharma, Shandian Zhe +2

Scientific Computing Institute, University of Utah, Salt Lake City, UT · Kahlert School of Computing, University of Utah, Salt Lake City, UT

PDE Operator LearningOperator LearningDeep Operator NetworksMixture-of-Experts ModelsFourier Neural Operator
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Aug 27, 2026

Enforcing Dirichlet Boundary Conditions in Operator Learning

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.

Andrew M. Stuart, Margaret Trautner

Caltech · ETH Zürich Department of Mathematics

PDE Operator LearningNeural OperatorsPDE SolvingNeural Network Approximation Theory
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Aug 12, 2026

Kernel Methods for Learning Operators with Multiple Inputs and Outputs

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

Adrien Weihs, Chunyang Liao, Jingmin Sun +1

Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, USA. · Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, 72701, USA. · Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218, USA.

Kernel MethodsPDE Operator LearningOperator Learning
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