cs.LGApr 10, 2026

Meta-Learned Basis Adaptation for Parametric Linear PDEs

Authors: Vikas Dwivedi, Monica Sigovan, Bruno Sixou

Organizations: CREATIS Biomedical Imaging Laboratory INSA, CNRS UMR 5220, Inserm, Université Lyon 1 Lyon 69621, France

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Abstract

We propose a hybrid physics-informed framework for solving families of parametric linear partial differential equations (PDEs) by combining a meta-learned predictor with a least-squares corrector. The predictor, termed KAPI (Kernel-Adaptive Physics-Informed meta-learner), is a shallow task-conditioned model that maps query coordinates and PDE parameters to solution values while internally generating an interpretable, task-adaptive Gaussian basis geometry. A lightweight meta-network maps PDE parameters to basis centers, widths, and activity patterns, thereby learning how the approximation space should adapt across the parametric family. This predictor-generated geometry is transferred to a second-stage corrector, which augments it with a background basis and computes the final solution through a one-shot physics-informed Extreme Learning Machine (PIELM)-style least-squares solve. We evaluate the method on four linear PDE families spanning diffusion, transport, mixed advection--diffusion, and variable-speed transport. Across these cases, the predictor captures meaningful physics through localized and transport-aligned basis placement, while the corrector further improves accuracy, often by one or more orders of magnitude. Comparisons with parametric PINNs, physics-informed DeepONet, and uniform-grid PIELM correctors highlight the value of predictor-guided basis adaptation as an interpretable and efficient strategy for parametric PDE solving.

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Figure 1: Overview of the proposed predictor-corrector framework. The full KAPI predictor is a shallow task-conditioned model that takes PDE parameters λ\lambda and a query point z\mathbf{z} and produces a coarse prediction upred(z;λ)u^{\mathrm{pred}}(\mathbf{z};\lambda) . Its task dependence is mediated through an internal parameter-conditioning network, which generates predictor basis structure including centers, widths, activity patterns, and other task-dependent basis parameters. The corrector then inherits the most informative predictor-guided basis elements, augments them with predictor-informed refinement and a background scaffold, and computes the final coefficients through a one-shot PIELM-style least-squares solve. The resulting corrected solution ucorr(z;λ)u^{\mathrm{corr}}(\mathbf{z};\lambda) combines amortized task-adaptive basis generation with physics-constrained coefficient recovery.
Figure 2: Comparison of exact and learned solution fields, together with absolute error maps, for the 2D Poisson problem. The left block shows the exact solution and the predictions of the KAPI predictor, FiLM-HyperPINN, and PI-DeepONet. The right block shows the corresponding absolute errors. The four rows correspond to the test cases (x0,y0,ν)=(0.50,0.50,0.07)(x_{0},y_{0},\nu)=(0.50,0.50,0.07) , (0.45,0.55,0.09)(0.45,0.55,0.09) , (0.30,0.30,0.06)(0.30,0.30,0.06) , and (0.50,0.50,0.03)(0.50,0.50,0.03) . The training range is x0∈[0.40,0.60]x_{0}\in[0.40,0.60] , y0∈[0.40,0.60]y_{0}\in[0.40,0.60] , and ν∈[0.05,0.10]\nu\in[0.05,0.10] .
Test case (x0,y0,ν)(x_{0},y_{0},\nu)Range typeKAPI predictorFiLM-HyperPINNPI-DeepONet
(0.50, 0.50, 0.07)(0.50,\,0.50,\,0.07)in-range2.008×10−22.008\times 10^{-2}1.657×10−21.657\times 10^{-2}2.946×10−22.946\times 10^{-2}
(0.45, 0.55, 0.09)(0.45,\,0.55,\,0.09)in-range1.195×10−21.195\times 10^{-2}6.055×10−36.055\times 10^{-3}1.345×10−21.345\times 10^{-2}
(0.30, 0.30, 0.06)(0.30,\,0.30,\,0.06)out-of-range3.569×10−13.569\times 10^{-1}2.717×10−12.717\times 10^{-1}4.129×10−14.129\times 10^{-1}
(0.50, 0.50, 0.03)(0.50,\,0.50,\,0.03)out-of-range3.869×10−23.869\times 10^{-2}9.953×10−29.953\times 10^{-2}1.543×10−11.543\times 10^{-1}
Table 1: Relative L2L_{2} errors for the 2D Poisson predictor on four representative test cases.
Figure 3: Comparison of exact and learned solution fields, together with absolute error maps, for the 1D linear advection problem. The left block shows the exact solution and the predictions of the KAPI predictor, FiLM-HyperPINN, and PI-DeepONet. The right block shows the corresponding absolute errors. The four rows correspond to the test cases (x0,ν)=(0.50,0.07)(x_{0},\nu)=(0.50,0.07) , (0.30,0.09)(0.30,0.09) , (0.65,0.04)(0.65,0.04) , and (0.50,0.02)(0.50,0.02) . The training range is x0∈[0.35,0.65]x_{0}\in[0.35,0.65] and ν∈[0.03,0.12]\nu\in[0.03,0.12] .
Test case (x0,ν)(x_{0},\nu)Range typeKAPI predictorFiLM-HyperPINNPI-DeepONet
(0.50, 0.07)(0.50,\,0.07)in-range4.022×10−24.022\times 10^{-2}5.581×10−25.581\times 10^{-2}2.950×10−12.950\times 10^{-1}
(0.65, 0.04)(0.65,\,0.04)in-range1.166×10−11.166\times 10^{-1}2.047×10−12.047\times 10^{-1}8.450×10−18.450\times 10^{-1}
(0.30, 0.09)(0.30,\,0.09)out-of-range2.215×10−22.215\times 10^{-2}4.203×10−24.203\times 10^{-2}1.715×10−11.715\times 10^{-1}
(0.50, 0.02)(0.50,\,0.02)out-of-range3.656×10−13.656\times 10^{-1}3.752×10−13.752\times 10^{-1}7.383×10−17.383\times 10^{-1}
Table 2: Relative L2L_{2} errors for the 1D linear advection predictor on four representative test cases.
Figure 4: Interpretability of the KAPI predictor through task-dependent basis geometry. In Poisson, the learned basis adapts to the source location and localization scale. In linear advection, the most active basis centers move along the dominant transport direction.
Figure 5: Predictor-corrector performance on the 2D Poisson family. Rows show the exact solution uexactu_{\mathrm{exact}} , the predictor output upredu_{\mathrm{pred}} , the corrected solution ucorru_{\mathrm{corr}} , and the corresponding absolute error maps ∣upred−uexact∣|u_{\mathrm{pred}}-u_{\mathrm{exact}}| and ∣ucorr−uexact∣|u_{\mathrm{corr}}-u_{\mathrm{exact}}| . Columns correspond to the test cases (x0,y0,ν)=(0.50,0.50,0.07)(x_{0},y_{0},\nu)=(0.50,0.50,0.07) , (0.45,0.55,0.09)(0.45,0.55,0.09) , (0.30,0.30,0.06)(0.30,0.30,0.06) , and (0.50,0.50,0.03)(0.50,0.50,0.03) . The first two cases lie within the training range, while the latter two probe extrapolation in source location and localization scale.
Figure 6: Predictor-corrector performance on the 1D periodic linear advection family. Rows show the exact solution uexactu_{\mathrm{exact}} , the predictor output upredu_{\mathrm{pred}} , the corrected solution ucorru_{\mathrm{corr}} , and the corresponding absolute error maps ∣upred−uexact∣|u_{\mathrm{pred}}-u_{\mathrm{exact}}| and ∣ucorr−uexact∣|u_{\mathrm{corr}}-u_{\mathrm{exact}}| . Columns correspond to the test cases (x0,ν)=(0.50,0.07)(x_{0},\nu)=(0.50,0.07) , (0.30,0.09)(0.30,0.09) , (0.65,0.04)(0.65,0.04) , and (0.50,0.02)(0.50,0.02) . The first three cases lie within the training range, while the last case probes extrapolation to a narrower packet than seen during training.
Figure 7: Predictor-corrector performance on the advection–diffusion family. Rows show the exact solution uexactu_{\mathrm{exact}} , the predictor output upredu_{\mathrm{pred}} , the corrected solution ucorru_{\mathrm{corr}} , and the corresponding absolute error maps ∣upred−uexact∣|u_{\mathrm{pred}}-u_{\mathrm{exact}}| and ∣ucorr−uexact∣|u_{\mathrm{corr}}-u_{\mathrm{exact}}| . Columns correspond to the test cases (a,ν)=(0.75,0.03)(a,\nu)=(0.75,0.03) , (0.55,0.045)(0.55,0.045) , (0.95,0.015)(0.95,0.015) , and (0.75,0.008)(0.75,0.008) . The first three cases lie within the training range, while the last case probes extrapolation to a lower-viscosity regime than seen during training.
Figure 8: Predictor-corrector performance on the variable-speed advection family. Rows show the exact solution uexactu_{\mathrm{exact}} , the predictor output upredu_{\mathrm{pred}} , the corrected solution ucorru_{\mathrm{corr}} , and the corresponding absolute error maps ∣upred−uexact∣|u_{\mathrm{pred}}-u_{\mathrm{exact}}| and ∣ucorr−uexact∣|u_{\mathrm{corr}}-u_{\mathrm{exact}}| . Columns correspond to the test cases (x0,ν,β)=(0.50,0.07,0.35)(x_{0},\nu,\beta)=(0.50,0.07,0.35) , (0.30,0.09,0.55)(0.30,0.09,0.55) , (0.65,0.04,0.45)(0.65,0.04,0.45) , and (0.50,0.04,0.70)(0.50,0.04,0.70) . The first three cases lie within the training range, while the last case probes extrapolation to a stronger velocity modulation than seen during training.
PDE familyTest caseRegimeEpred\mathcal{E}_{\mathrm{pred}}Ecorr\mathcal{E}_{\mathrm{corr}}
2D Poisson(0.50, 0.50, 0.070)(0.50,\,0.50,\,0.070)in-range2.008×10−22.008\times 10^{-2}7.032×10−47.032\times 10^{-4}
 (0.45, 0.55, 0.090)(0.45,\,0.55,\,0.090)in-range1.195×10−21.195\times 10^{-2}5.596×10−45.596\times 10^{-4}
 (0.30, 0.30, 0.060)(0.30,\,0.30,\,0.060)out-of-range3.569×10−13.569\times 10^{-1}1.122×10−31.122\times 10^{-3}
 (0.50, 0.50, 0.030)(0.50,\,0.50,\,0.030)out-of-range3.869×10−23.869\times 10^{-2}1.656×10−31.656\times 10^{-3}
Linear advection(0.50, 0.070)(0.50,\,0.070)in-range4.022×10−24.022\times 10^{-2}3.428×10−23.428\times 10^{-2}
 (0.30, 0.090)(0.30,\,0.090)in-range2.215×10−22.215\times 10^{-2}2.040×10−22.040\times 10^{-2}
Table 3: Predictor and corrector errors across the four PDE families. The corrector typically reduces the error by one or more orders of magnitude, including several extrapolative cases. The main exception is constant-coefficient linear advection, where the predictor already captures the dominant transport geometry and the correction is correspondingly modest.
Figure 9: Interpretability of the predictor-corrector mechanism through basis geometry. In each panel, the top row shows the predictor geometry, while the bottom row shows the enriched corrector geometry. Marker size indicates basis width, and color indicates normalized coefficient magnitude. The predictor supplies a task-adaptive concentration of basis support near the important region, while the corrector augments this inherited basis with additional refinement and background support, and then recomputes the final coefficients through least squares.
PDE familyTest caseRegimeGuided correctorUniform-only corrector
2D Poisson(0.50, 0.50, 0.070)(0.50,\,0.50,\,0.070)in-range7.032×10−47.032\times 10^{-4}1.138×10−21.138\times 10^{-2}
 (0.45, 0.55, 0.090)(0.45,\,0.55,\,0.090)in-range5.596×10−45.596\times 10^{-4}2.659×10−22.659\times 10^{-2}
 (0.30, 0.30, 0.060)(0.30,\,0.30,\,0.060)out-of-range1.122×10−31.122\times 10^{-3}4.425×10−24.425\times 10^{-2}
 (0.50, 0.50, 0.030)(0.50,\,0.50,\,0.030)out-of-range1.656×10−31.656\times 10^{-3}3.445×10−13.445\times 10^{-1}
Advection– diffusion(0.75, 0.030)(0.75,\,0.030)in-range1.959×10−41.959\times 10^{-4}1.126×10−21.126\times 10^{-2}
 (0.55, 0.045)(0.55,\,0.045)in-range2.225×10−42.225\times 10^{-4}3.640×10−33.640\times 10^{-3}
Table 4: Ablation study comparing the predictor-guided corrector against a uniform-only PIELM corrector. For each case, the uniform-only result reports the best error obtained over the tested background-grid sweep. Across both PDE families, the predictor-guided corrector is consistently more accurate, often by one to three orders of magnitude.
Figure 10: Ablation study on the 2D Poisson family. We compare the predictor-guided corrector against a uniform-only PIELM corrector as the background grid side length is varied. Across all four cases, the predictor-guided corrector remains substantially more accurate, and the gap is especially large in the shifted and narrow out-of-range settings. This shows that increasing uniform-grid density alone does not recover the benefit of predictor-guided basis placement.
Figure 11: Ablation study on the advection–diffusion family. We compare the predictor-guided corrector against a uniform-only PIELM corrector as the background space-time RBF count is varied. The predictor-guided corrector consistently achieves much lower error, particularly in the transport-dominated and low-viscosity regimes. Thus, the main gain comes from task-adaptive basis geometry rather than from least-squares correction alone.
PDEMethodRel. L2L^{2} errorTraining time (s)Inference time (s)Retraining per new task
PoissonKAPI predictor1.294×10−21.294\times 10^{-2}231.19231.191.843×10−31.843\times 10^{-3}No
PoissonSingle-instance PINN5.099×10−25.099\times 10^{-2}58.2358.231.132×10−31.132\times 10^{-3}Yes
AdvectionKAPI predictor4.022×10−24.022\times 10^{-2}369.38369.384.026×10−24.026\times 10^{-2}No
AdvectionSingle-instance PINN2.897×10−22.897\times 10^{-2}150.96150.962.578×10−22.578\times 10^{-2}Yes
Table 5: Ablation comparing the KAPI predictor against shallow single-instance PINNs on one representative Poisson task and one representative linear-advection task. KAPI incurs a larger offline training cost because it is trained over an entire parameter family, but it does not require retraining for each new PDE instance.
Figure 12: Poisson ablation on the fixed task (x0,y0,ν)=(0.50,0.50,0.07)(x_{0},y_{0},\nu)=(0.50,0.50,0.07) . Top row: exact finite-difference solution, KAPI prediction, and single-instance PINN prediction. Bottom row: corresponding absolute error maps. The KAPI predictor yields lower error on this representative instance, while the single-instance PINN exhibits a broader and more diffuse error pattern.
Figure 13: Training curves for the Poisson ablation. The KAPI loss is noisier because each update is computed over a batch of different tasks from the parametric family, whereas the single-instance PINN optimizes a fixed one-task objective.
Figure 14: Advection ablation on the fixed task (x0,ν)=(0.50,0.07)(x_{0},\nu)=(0.50,0.07) . Top row: exact solution, KAPI prediction, and single-instance PINN prediction. Bottom row: corresponding absolute error maps. Both methods capture the transported ridge accurately, while the single-instance PINN is slightly sharper on this fixed instance.
Figure 15: Training curves for the linear-advection ablation. The KAPI trajectory is noisier because each update is computed across a batch of different advection tasks, whereas the single-instance PINN optimizes only one fixed PDE objective.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

CaseKernelsKAPI predictorMain training settingsCorrector
PoissonM=128M=1282-layer internal parameter-conditioning network, width 642000 epochs; Adam; lr =10−3=10^{-3} ; 4 tasks/batch; mixed localized and uniform interior samplingSelected predictor-guided spatial basis with local enrichment and background support; ridge LS with weak anchor term
Linear advectionM=32M=32Internal task encoder width 64; dynamic net width 96; time Fourier features5000 epochs; Adam; lr =10−3=10^{-3} ; 4 tasks/batch; interior + IC + periodic BC + near-IC samplesSelected predictor-guided spacetime basis with local refinement and periodic background; ridge LS
Advection–diffusionM=48M=483-layer dynamic predictor, width 128; time Fourier featuresAdam; lr =10−3=10^{-3} ; uniform + localized interior samples; IC and BC samples; viscosity curriculumPredictor-guided spacetime basis with refinement centers and background support; ridge LS with λridge=10−8\lambda_{\mathrm{ridge}}=10^{-8}
Variable-speed advectionM=64M=64Dynamic predictor, width 128; periodic base dictionary5000 epochs; Adam; lr =5×10−4=5\times 10^{-4} ; uniform + localized interior samples; IC and periodic BC samplesPredictor-guided spacetime basis with gradient-informed refinement and background support; ridge LS with λridge=10−8\lambda_{\mathrm{ridge}}=10^{-8}
Appendix
Table 6: Main implementation settings used across the four PDE families. We report only the principal architectural and training choices; full code and lower-level details will be released.
Figure 16: Two-source extension of the 2D Poisson predictor-corrector experiment. Each column shows one test case; rows show the reference solution, predictor, corrected solution, predictor error, and corrector error.
Figure 17: Four-source extension of the 2D Poisson predictor-corrector experiment. Each column shows one test case; rows show the reference solution, predictor, corrected solution, predictor error, and corrector error.
Figure 18: Predictor-guided corrector geometries for the multi-source Poisson extensions.
Test case (x0,σ)(x_{0},\sigma)Epred\mathcal{E}_{\mathrm{pred}}Ecorr\mathcal{E}_{\mathrm{corr}}
(0.50, 0.070)(0.50,\,0.070)8.541×10−38.541\times 10^{-3}8.398×10−38.398\times 10^{-3}
(0.35, 0.050)(0.35,\,0.050)1.039×10−21.039\times 10^{-2}1.027×10−21.027\times 10^{-2}
(0.68, 0.090)(0.68,\,0.090)5.903×10−35.903\times 10^{-3}5.677×10−35.677\times 10^{-3}
(0.50, 0.040)(0.50,\,0.040)1.105×10−21.105\times 10^{-2}1.100×10−21.100\times 10^{-2}
Appendix
Table 7: Relative L2L^{2} errors for the Mexican-hat advection test cases.
Figure 19: Predictor-corrector performance for the Mexican-hat advection test. Rows show uexactu_{\mathrm{exact}} , upredu_{\mathrm{pred}} , ucorru_{\mathrm{corr}} , ∣upred−uexact∣|u_{\mathrm{pred}}-u_{\mathrm{exact}}| , and ∣ucorr−uexact∣|u_{\mathrm{corr}}-u_{\mathrm{exact}}| .
Figure 20: Time-slice comparison for the Mexican-hat advection test. Solid blue curves show ucorru_{\mathrm{corr}} and dashed red curves show uexactu_{\mathrm{exact}} at t=0,0.25,0.5,0.75,1.0t=0,0.25,0.5,0.75,1.0 .
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