Learning PDE Dynamics between Submanifolds Using Green's Observation Operators

Jan Tauberschmidt; Jephte Abijuru; Samuel Okon; Naukshatro Bose; Sophie Fellenz; Marius Kloft; Jonas Latz; Sebastian Josef Vollmer

cs.LGOct 1, 2026

Learning PDE Dynamics between Submanifolds Using Green's Observation Operators

Authors: Jan Tauberschmidt, Jephte Abijuru, Samuel Okon, Naukshatro Bose, Sophie Fellenz, Marius Kloft, Jonas Latz, Sebastian Josef Vollmer

Organizations: German Research Center for Artificial Intelligence (DFKI), Kaiserslautern, Germany · Rheinland-Pfälzische Technische Universität (RPTU), Kaiserslautern, Germany · University of Manchester, Manchester, United Kingdom

Abstract

Many physical systems are driven and observed only on lower-dimensional submanifolds of a larger spatial domain, while their dynamics are governed by the ambient medium occupying that domain. Examples include laser-heated parts imaged by an infrared camera, and ground-level emissions measured on a sensor plane. Full-domain solvers, however, compute the entire volume for every new source although only the observation submanifold is needed, and black-box surrogates do not exploit that the ambient medium remains fixed. We introduce the \emph{Green's Observation Operator (GObO)}, which maps the ambient medium once to the Green's kernel of a linear PDE restricted to the source and observation submanifolds. New sources then cost one lower-dimensional integral and no network evaluation. Exponential rates in the kernel yield an exact finite streaming state with horizon-independent memory; we prove its stability and an approximation rate for the restricted heat kernel. On three-dimensional heat conduction and advection--diffusion with collocated and distinct source and observation geometries, GObO trained on static sources predicts responses to moving sources zero-shot with 4--8 $\times$ lower error than black-box surrogates, at 1.4,ms per query after a single conditioning pass. The same kernel transfers across resolutions and admits corrections for mild nonlinearities, including radiative losses and temperature-dependent conductivity, without retraining, at the cost of lower in-distribution accuracy.

Figures & tables

Figure 1: A laser acts on a source submanifold and drives an observed temperature response on the observation submanifold. The ambient medium affects the temperature flux. Classical PDE solvers model the transport through the full volume, whereas our GObO directly models the map between source and observation submanifolds.

	Ours	Finite-volume solver
Time for 1000 source predictions/solves	$\mathbf{1.7}$ s	$1.3$ h

Table 1: Classical solvers repeat a volumetric solve for every source although only the lower-dimensional trace is required, making repeated prediction expensive. See details in Section 5.5 .

system	method	static	moving
heat	GObO	$0.011\pm 0.001$	$\mathbf{0.012\pm 0.000}$
	adapted NGO, $S=1$	$0.039\pm 0.001$	$0.043\pm 0.000$
	medium-conditioned FNO-2D	$\mathbf{0.001\pm 0.000}$	$0.101\pm 0.004$
advection–diffusion	GObO, fixed rates	$0.114\pm 0.018$	$\mathbf{0.101\pm 0.020}$
	medium-conditioned FNO-2D	$\mathbf{0.104\pm 0.009}$	$0.430\pm 0.015$

Table 2: GObO transfers to unseen source configurations. All models are trained on static sources and evaluated on static and moving sources without retraining. Shown is the mean relative error and standard deviation.

Figure 2: GObO tracks the moving response, whereas FNO does not track it accurately. Left: true final temperature and source trajectory. Middle: temperature profiles at the three indicated times. Right: position and magnitude of the temperature peak.

Panel of Figure 3 — Figure 3: The restricted observation requires additional state information for recurrent prediction. (a) Next-step regression error with increasing observation history, compared with a closed two-dimensional heat system. (b) Rollout error of GObO and transient NGO with spectral cap $S$ . (c) History dependence of the purely spatial one-step GObO variant.

method	frequency	CPU	GPU, $b=1$
FV solver, matched accuracy	per source	$4.6$ s	n/a
FNO-2D forward	per source	$194.1$ ms	$2.50$ ms
GObO conditioning	per medium	$218.3$ ms	$10.1$ ms
GObO query	per source	$\mathbf{1.45}$ ms	$\mathbf{0.45}$ ms

Table 3: GObO constructs one kernel per medium and then evaluates new sources faster than FNO-2D or the numerical solver.

Appendix figures & tables38 assets

Supplementary material from the paper’s appendix.

Appendix

	pointwise kernel values	reusable representation
steady	GreenLearning, MOD-Net, BI-GreenNet, and GreensONet ( Boullé et al., 2022a ; Zhang et al., 2022 ; Lin et al., 2023 ; Gu et al., 2025 )	discrete and spectral Green representations ( Tang et al., 2022 ; Praveen et al., 2025 ; Yoo et al., 2026 ; Yang et al., 2026 )
transient	GreenONet ( Aldirany et al., 2024 )	full-kernel identification ( Boullé et al., 2022b ) , one-step NGO, DGNet, and GObO

Appendix

Table 4: Representative learned Green-function methods. Pointwise methods query individual kernel values. Reusable methods produce a finite operator representation that is applied without another network query at each point.

quantity	value
spatial grid	$64^{3}$ cells
time interval	$[0,1]$
stored frames	$N_{t}=64$
time integrator	Crank–Nicolson
substeps per frame	$4$ for linear, $32$ for radiative, adaptive for configuration B
linear solver	preconditioned conjugate gradients

Appendix

Table 5: Reference discretization for the heat data.

medium	construction
two-phase	Threshold $g_{\mathrm{med}}$ at its median. Set $\alpha_{\mathrm{solid}}=5\times 10^{-4}$ and $\alpha_{\mathrm{powder}}=5\times 10^{-6}$ . The top 2 cell layers are set to $\alpha_{\mathrm{powder}}$ .
graded	Set $\varphi=\Phi(g_{\mathrm{med}}/\operatorname{std}(g_{\mathrm{med}}))$ and $\alpha=\alpha_{\mathrm{powder}}+(\alpha_{\mathrm{solid}}-\alpha_{\mathrm{powder}})\varphi$ . For the interior-plane data, the endpoints are $10^{-4}$ and $10^{-2}$ and no powder cap is used.

Appendix

Table 6: Media used in the heat experiments. The collocated experiments use the two-phase medium. The interior-plane experiments use the graded medium.

data set	samples
heat training	$1024$
static test	$128$
straight scans	$128$
serpentine scans	$128$
radiative test	$128$
Kirchhoff data	$1024/128/128$ per setting

Appendix

Table 7: Heat data used for training and evaluation. Multiple numbers denote training, static test, and the moving or shifted test sets.

observation	grid location	relation to the source
horizontal plane	$i_{z}=48$	separated by about $L_{\Omega}/4$
vertical plane	$i_{x}=32$	mid-plane whose top edge lies in the source face

Appendix

Table 8: Interior observation surfaces. Both cases are covered by Proposition 2 . The horizontal case also satisfies the uniform bound in Remark 1 .

data set	samples
training	$1024$
static test	$128$
straight moving source	$128$
complex moving source	$128$

Appendix

Table 9: Advection–diffusion data used for training and evaluation.

quantity	setting
retained modes	$32\times 32$ in each chart
source and observation dimensions	$D_{S}=D_{O}=1024$
temporal rates	$R=16$
factor rank per rate	$L=16$
hidden width	$64$
conditioning blocks	4 cosine spectral blocks with U-Nets and 1 truncated layer

Appendix

Table 10: Settings for the heat GObO used in the main experiments.

Figure 4: Conditioning architectures. The adapted NGO system network maps medium projections to a dense coefficient matrix. GObO maps the medium volume to low-rank factors for each analytic decay rate. The GObO source is applied only after these factors have been produced.

quantity	setting
loss	relative $L_{2}$ in equation 108
optimizer	AdamW with default moment parameters
learning rate	$10^{-3}$
weight decay	$10^{-4}$
schedule	cosine decay to zero
training steps	$30{,}000$

Appendix

Table 11: Shared training settings. FNO-3D uses batch size 8. The interior-plane models use the changes described below.

quantity	value
diffusivity	$\kappa=1$
time step and frames	$\Delta t=0.0125$ and $N_{t}=64$
first supervised frame	$4$ , corresponding to $\tau_{\min}=0.05$
chart grid	$64\times 64$ cell-centered points
retained chart modes	$16\times 16$ cosine modes
ambient spectral cutoff	$n_{j}\leq 6$ , giving $343$ modes

Appendix

Table 12: Settings for the controlled curved-manifold example.

quantity	setting
CPU	AMD EPYC 7742 with 32 allocated cores
GPU	NVIDIA A100-SXM4 with 80 GB
software	Python 3.12.3 and PyTorch 2.13
numerical libraries	NumPy 2.1.0 and SciPy 1.17.1 with OpenBLAS
solver repetitions	1 warmup and 3 measurements
learned-model repetitions	3 warmups and 8 measurements

Appendix

Table 13: Environment used for the timing measurements.

mode	computation
M0	Compare the 2 numerical reference solutions.
M1	Run the complete learned model on the fine grid.
M2	Run the learned model on the coarse grid and evaluate its final continuous spectral representation on the fine grid.
M3	Run only the GObO medium encoder on the coarse grid, then project the fine-grid source and evaluate the operator on the fine grid.

Appendix

Table 14: Resolution-transfer modes used in Section F.5.2 .

setting	$\beta u_{\max}$	conductivity ratio
mild	$0.2$	$1.2$
strong	$0.5$	$1.5$
severe	$1.0$	$2.0$

Appendix

Table 15: Conductivity contrasts used in the nonlinear experiments. The range covers reported process windows for 316L, IN718, and Ti-6Al-4V ( Kim, 1975 ; Alloys, 2008 ; Mukherjee et al., 2016 ) .

model	parameters	scope
GObO	$22.16$ M	heat, fixed rates
FNO-2D	$16.91$ M	heat
matched spectral model	$17.89$ M	heat
FNO-3D	$67.18$ M	heat
advection GObO, fixed rates	$44.01$ M	advection–diffusion
advection GObO, learned rates	$44.14$ M	advection–diffusion

Appendix

Table 16: Model sizes. Fixed-rate and learned-rate advection models are listed separately.

	relative $L_{2}$			absolute $L_{2}$
method	static	straight	serpentine	static	straight	serpentine
GObO	$0.011\pm 0.001$	$0.012\pm 0.000$	$0.005\pm 0.000$	$2838\pm 160$	$2253\pm 85$	$2447\pm 12$
FNO-2D	$0.001\pm 0.000$	$0.101\pm 0.004$	$0.137\pm 0.007$	$282\pm 17$	$23330\pm 810$	$63590\pm 3400$
FNO-3D	$0.004\pm 0.000$	$0.185\pm 0.003$	$0.426\pm 0.008$	$970\pm 6$	$46670\pm 360$	$199500\pm 3900$
matched spectral model	$0.003\pm 0.001$	$0.204\pm 0.002$	$0.217\pm 0.004$	$370\pm 50$	$48130\pm 700$	$100800\pm 1900$

Appendix

Table 17: Complete source-transfer comparison. Values are means and population standard deviations over 3 seeds. The moving sets are zero-shot.

Panel of Figure 5 — Figure 5: Per-frame behavior on serpentine scans. GObO tracks the moving peak throughout the trajectory, whereas FNO-2D develops a persistent displacement error. Curves show the 3-seed min–max band.

baseline architecture	static	straight	serpentine
FNO-2D, 4 layers	$0.001\pm 0.000$	$0.101\pm 0.004$	$0.137\pm 0.007$
FNO-2D, 5 layers	$0.001\pm 0.000$	$0.103\pm 0.002$	$0.149\pm 0.005$
matched spectral model, 4 layers	$0.003\pm 0.001$	$0.204\pm 0.002$	$0.217\pm 0.004$
matched spectral model, 5 layers	$0.004\pm 0.001$	$0.226\pm 0.002$	$0.234\pm 0.013$

Appendix

Table 18: Baseline-depth ablation (relative

L_{2}

). Increasing depth does not resolve the moving-source failure of either direct-output architecture.

GObO factorization	static	straight	serpentine
fixed rates, shared factors, $L=16$	$\mathbf{0.011\pm 0.001}$	$\mathbf{0.012\pm 0.000}$	$\mathbf{0.005\pm 0.000}$
learned rates, shared factors, $L=16$	$0.011\pm 0.002$	$0.012\pm 0.002$	$0.006\pm 0.001$
fixed rates, separate factors, $L=16$	$0.019\pm 0.001$	$0.016\pm 0.001$	$0.008\pm 0.000$
learned rates, separate factors, $L=16$	$0.013\pm 0.002$	$0.013\pm 0.002$	$0.006\pm 0.001$
fixed rates, shared factors, $L=1$	$0.130\pm 0.007$	$0.123\pm 0.005$	$0.074\pm 0.008$

Appendix

Table 19: Rate and factor-structure ablation. Here

L

is the factor search rank used for each fixed or learned rate, matching the terminology of Tables 26 and 26 .

observation	method	static	straight	serpentine
horizontal plane	GObO	$0.042$	$\mathbf{0.044}$	$\mathbf{0.027}$
horizontal plane	FNO-2D	$\mathbf{0.027}$	$0.194$	$0.108$
vertical mid-plane	GObO	$0.064$	$\mathbf{0.062}$	$\mathbf{0.054}$
vertical mid-plane	FNO-2D	$0.066$	$0.240$	$0.148$

Appendix

Table 20: Source transfer with non-collocated observations. The horizontal plane is separated from the source by

L_{\Omega}/4

, the vertical plane is orthogonal to the source face.

Panel of Figure 6 — Figure 6: A moving source observed on the vertical mid-plane. For the displayed sample, GObO has relative error $0.033$ and FNO-2D has $0.104$ , the aggregate per-frame curves show that the gap persists over the test set.

	relative $L_{2}$			absolute $L_{2}$
method	static	straight	complex	static	straight	complex
GObO, fixed rates	$0.114\pm 0.018$	$0.101\pm 0.020$	$0.139\pm 0.035$	$1.88\pm 0.19$	$2.04\pm 0.31$	$2.00\pm 0.24$
GObO, learned rates	$0.300\pm 0.021$	$0.313\pm 0.016$	$0.352\pm 0.028$	$5.19\pm 0.19$	$6.68\pm 0.23$	$6.13\pm 0.26$
FNO-2D	$0.104\pm 0.009$	$0.430\pm 0.015$	$0.364\pm 0.007$	$1.77\pm 0.15$	$9.33\pm 0.36$	$6.09\pm 0.17$

Appendix

Table 21: Complete source-generalization results on the sensor curtain. Values are means and population standard deviations over 3 seeds.

Figure 7: A moving ground-level release observed on a vertical sensor curtain. The qualitative comparison complements the aggregate advection–diffusion errors in Table 21 .

	params	one-step (static)	rollout, static	straight	serpentine	rescaled fraction	ms/frame
GObO	22.16 M	–	$0.011\pm 0.001$	$0.012\pm 0.000$	$0.005\pm 0.000$	–	0.13
A. NGO as published
NGO, $S=0.8$	0.02 M	$0.163\pm 0.000$	$0.721\pm 0.000$	$0.760\pm 0.000$	$0.552\pm 0.001$	$1.00$	0.09
NGO, $S=1$	0.02 M	$0.012\pm 0.000$	$0.039\pm 0.001$	$0.043\pm 0.000$	$0.027\pm 0.001$	$1.00$	0.09
B. One-step GObO variants
one-step, $k=1$	22.28 M	$0.021\pm 0.000$	$0.103\pm 0.003$	$0.103\pm 0.009$	$0.058\pm 0.003$	$1.00$	0.08

Appendix

Table 22: One-step schemes on the trace. Relative error, mean

\pm

population standard deviation over 3 seeds, 128 samples per set. “One-step” is the error of a single step from the true previous frames, the quantity these rows are trained on, the 3 rollout columns feed the model its own previous frames from rest. “Rescaled fraction” indicates whether the norm cap is active on the first static test sample, averaged over the 3 seeds. Frame-spacing transfer requires retraining for every one-step row. Parameter counts are of the built models.

cumulative error on	$[0,1T]$	$[0,2T]$	$[0,3T]$	$[0,4T]$
GObO	$0.010\pm 0.000$	$0.047\pm 0.002$	$0.105\pm 0.004$	$0.158\pm 0.007$
NGO, $S=0.8$	$0.702\pm 0.000$	$0.812\pm 0.000$	$0.832\pm 0.000$	$0.841\pm 0.000$
NGO, $S=1$	$0.036\pm 0.001$	$0.142\pm 0.001$	$0.237\pm 0.001$	$0.297\pm 0.001$
one-step, $k=1$	$0.099\pm 0.001$	$0.227\pm 0.005$	$0.326\pm 0.008$	$0.396\pm 0.012$
one-step, $k=4$	$0.090\pm 0.012$	diverged	diverged	diverged
one-step, $k=16$	$0.030\pm 0.006$	$0.136\pm 0.022$	$0.58\pm 0.20$	diverged

Appendix

Table 23: One-step schemes, horizon. Cumulative relative error on the first

h\cdot 64

frames of the decaying-source test set (128 samples, source off before

T

), mean

\pm

population standard deviation over 3 seeds. GObO by the streaming recurrence, the one-step rows by rollout. “Diverged” is a rollout that has left the scale of the data by that horizon (a cumulative error above

1

in at least 1 seed). The

S=0.8

row has decayed to a fraction of the truth instead.

Panel of Figure 8 — Figure 8: Impulse-response comparison. The panels show the response produced by the proposed kernel parametrization and by the naive separable space–time alternative.

	params	static	straight	serpentine	$\Delta/2$	$\Delta$ (control)	$2\Delta$
GObO ( $e^{-\lambda_{k}(t-s)}$ , 16 rates)	22.16 M	$0.011\pm 0.001$	$0.012\pm 0.000$	$0.005\pm 0.000$	$0.010\pm 0.000$	$0.011\pm 0.000$	$0.011\pm 0.000$
separable space–time kernel, $C=16$	22.67 M	$0.035\pm 0.001$	$0.045\pm 0.002$	$0.051\pm 0.008$	$0.031\pm 0.001$	$0.031\pm 0.001$	$0.033\pm 0.001$

Appendix

Table 24: Separable space–time basis. Relative error equation 108 , mean

\pm

population standard deviation over 3 seeds, 128 samples per set. The last 3 columns use the same independently generated frame-spacing test set.

	params	static	straight	serpentine	$\Delta/2$	$\Delta$ (control)	$2\Delta$	ms/frame
GObO (cosine spectral conditioning)	22.16 M	$0.011\pm 0.001$	$0.012\pm 0.000$	$0.005\pm 0.000$	$0.010\pm 0.000$	$0.011\pm 0.000$	$0.011\pm 0.000$	0.13
medium projections $+$ U-Net	22.73 M	$0.103\pm 0.012$	$0.102\pm 0.011$	$0.068\pm 0.011$	$0.110\pm 0.013$	$0.110\pm 0.013$	$0.111\pm 0.013$	0.14

Appendix

Table 25: Medium pathway. Relative error, mean

\pm

population standard deviation over 3 seeds, 128 samples per set. Columns as in Table 24 , ms/frame is measured per streaming step.

Panel of Figure 9 — Figure 9: Kernel recovery on distinct curved manifolds. The source lies on a saddle and the response is observed on a separate curved surface. The final panels compare the reference and recovered restricted kernel at $\tau=0.05$ .

search rank $L$	objective	recovered rank	coefficient error
coefficient optimum	$1.804\times 10^{-4}$	$7$	n/a
$1$	$2.593\times 10^{-3}$	$1$	$0.986$
$2$	$8.714\times 10^{-4}$	$2$	$0.730$
$4$	$2.563\times 10^{-4}$	$4$	$0.380$
$8$	$1.826\times 10^{-4}$	$7$	$0.215$
$16$	$1.788\times 10^{-4}$	$7$	$0.160$

Appendix

Table 26: Complete search-rank sweep. Recovered rank uses the threshold

10^{-4}\sigma_{1}

, coefficient error is

\lVert A-A_{\mathrm{cvx}}\rVert_{F}/\lVert A_{\mathrm{cvx}}\rVert_{F}

. The objective reaches its elbow at

L=8

, and

L=16

is used in the heat experiments. At

L=256

, the additional singular values lie at the reporting threshold, making the rank count threshold-sensitive.

operation	frequency	CPU	GPU $b=1$	GPU $b=64$ /sample
FV assembly and preconditioner	per medium	$0.6$ s	n/a	n/a
FV production trajectory	per source	$63.5$ s	n/a	n/a
FV matched-accuracy trajectory	per source	$4.6$ s	n/a	n/a
FNO-2D forward pass	per source	$194.1$ ms	$2.50$ ms	$0.38$ ms
GObO conditioning	per medium	$218.3$ ms	$10.1$ ms	$4.29$ ms
GObO batch response	per source	$1.45$ ms	$0.45$ ms	$0.008$ ms

Appendix

Table 27: Operation-level timing. A batch of 64 sources shares one conditioned GObO, the finite-volume solver was evaluated on CPU only.

method	time	memory
finite-volume solver	$O(N_{t}n_{\mathrm{sub}}I_{\mathrm{CG}}N^{3}\log N)$	$O(N^{3})$
FNO-2D	$O(PN^{2}(C^{2}+C\log N))$	$O(N^{2}C)$
GObO conditioning	$O(N^{3})$	operator factors
GObO batch	$O(N_{t}[N^{2}\log N+RL(D_{S}+D_{O})]+N_{t}^{2}RL)$	$O(N_{t}N^{2}+N_{t}^{2}R)$
GObO streaming	$O(N_{t}[N^{2}\log N+RL(D_{S}+D_{O})])$	$O(N^{2}+RL)$

Appendix

Table 28: Implemented asymptotic costs. Streaming removes the quadratic dependence on history length.

mode	GObO	FNO-2D
M0. compare truths at 64 and 128	$0.026$
M1. end-to-end at 128	$0.078\pm 0.022$	$0.044\pm 0.002$
M2. coarse compute, fine readout	$\mathbf{0.025\pm 0.000}$	$0.059\pm 0.000$
M3. condition at 64, operator at 128	$\mathbf{0.025\pm 0.000}$	n/a

Appendix

Table 29: Resolution transfer

64\to 128

(relative

L_{2}

against truth at 128, 3 seeds, 64 paired samples). M2 and M3 reach the discretization floor, M3 is structurally unavailable to FNO.

Figure 10: Resolution-transfer paths. The source enters FNO before its learned layers, so there is no FNO analogue of the GObO-only M3 path.

M1 positional encoding	GObO	FNO-2D
raw cell-index features	$0.292\pm 0.056$	$0.047\pm 0.001$
continuous reference features	$0.078\pm 0.022$	$0.044\pm 0.002$

Appendix

Table 30: Positional-encoding ablation for end-to-end fine evaluation. M2 and M3 do not run the encoder on the fine grid and are unaffected by this choice.

Panel of Figure 11 — Figure 11: Radiative feedback with the original linear checkpoint. (a) Peak surface temperature over time, the shaded band between the unchanged linear kernel and the truth is the heat that radiation removes. (b) Temperature at $t=T$ along the hottest row. The unchanged kernel overshoots the peak, whereas the streamed semi-implicit correction captures both saturation and the final profile without retraining.

Configuration A. exact Kirchhoff transform
		mild $0.2$		strong $0.5$		severe $1.0$
method	retraining	static	shifted	static	shifted	static	shifted
FNO trained for each setting	6	$\mathbf{0.001}$	$0.104$	$\mathbf{0.001}$	$0.116$	$\mathbf{0.001}$	$0.103$
linear FNO + transform	0	$\mathbf{0.001}$	$0.096$	$0.002$	$0.107$	$0.002$	$0.099$
GObO, no correction	0	$0.053$	$0.069$	$0.122$	$0.151$	$0.232$	$0.263$
GObO + exact transform	0	$0.011$	$\mathbf{0.009}$	$0.010$	$\mathbf{0.009}$	$0.010$	$\mathbf{0.009}$

Appendix

Table 31: Temperature-dependent conduction without retraining. Relative errors are means over 3 seeds. Every standard deviation is at most

0.002

for the GObO rows and at most

0.007

for the retrained FNO. Shifted sources include scans, rasters, and sums of 2 pulses.

Panel of Figure 12 — Figure 12: 2 nonlinear material laws corrected with 1 linear checkpoint. Surface temperature along the hottest row at $t=T$ for the severe-contrast tier ( $k_{\max}/k_{\min}=2$ ), 1 static test sample per configuration. Each panel overlays the truth, the uncorrected GObO prediction, and the corrected prediction. Panel (a) uses the exact Kirchhoff transform and panel (b) uses the trace-based state update. Both panels share axes, line styles, and temperature scale. Table 31 quantifies the contrast trends.

model	residual/state	steps	mild static	mild shifted	strong static	strong shifted
FNO trained for setting	n/a	0	$0.001$	$0.101$	$0.001$	$0.104$
linear FNO + iteration	time derivative	$\leq 20$	$0.011\,(2)$	$0.097\,(29)$	$0.026\,(6)$	$0.093\,(26)$
linear GObO + iteration	time derivative	$\leq 20$	$0.011\,(3)$	$\mathbf{0.009}\,(2)$	$0.015\,(3)$	$\mathbf{0.014}\,(3)$
linear GObO + iteration	generator	$\leq 20$	$0.011\,(3)$	$\mathbf{0.009}\,(2)$	$0.016\,(4)$	$\mathbf{0.014}\,(3)$
GObO + trace-based update	generator state	1	$0.014$	$0.012$	$0.021$	$0.019$
FV solver + iteration	time derivative	$\leq 20$	$0.002\,(1)$	n/a	$0.007\,(3)$	n/a

Appendix

Table 32: Fixed-point attribution for configuration B. Entries are median relative errors over non-diverged samples, parentheses give half the interquartile range in units of the last digit.

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Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension

Rüdiger Kempf

Applied and Numerical Analysis University of Bayreuth 95440 Bayreuth, Germany

Probabilistic Operator Learning Physics-Informed Learning Token Budget Allocation

Open paper AskPDF arXiv

We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations. Our main theoretical contribution is an explicit budget allocation condition relating the number

N

of training pairs, the number

n

of input observations, and the output resolution

m

. The condition is derived from a coupled error analysis that interprets the surrogate as a reconstruction from approximate data. This yields a decomposition of the total error into reconstruction and learning contributions that can be analyzed independently. As a consequence, we obtain quantitative scaling laws describing how

N

n

, and

m

must be coupled to guarantee convergence and to balance offline learning and online reconstruction errors. The resulting estimates extend previous analyses of kernel-based operator learning. We further introduce a physics-informed extension that incorporates knowledge of the underlying PDE at evaluation time. Rather than encoding constraints directly into the kernel, we augment the online reconstruction step by penalizing PDE residuals at collocation points. The method requires no retraining for new inputs. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the proposed physics-informed reconstruction strategy.

Abstract

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