Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes

Jingye Li; Jinglai Li

cs.LGOct 28, 2026

Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes

Authors: Jingye Li, Jinglai Li

Organizations: School of Mathematics, University of Birmingham, UK

Abstract

In variable-step operator learning, agreement at observed step sizes does not by itself determine predictions at other temporal resolutions. We propose an operator whose step-size dependence is prescribed by a first-order Duhamel approximation of semilinear dynamics. It combines exact linear propagation with a local nonlinear function of the field and its derivatives, allowing evaluation below the observation interval without supervision densely sampled across step sizes. Under suitable conditions and at fixed spatial resolution, we prove that agreement between models at two step sizes controls their prediction differences throughout a prescribed range. We also give conditions for local identifiability of both components from two step sizes, supporting interpretation of the learned structure. A latent extension applies the same construction to learned fields. We evaluate the base model on two synthetic problems and the extension on two real datasets against several baselines.

Figures & tables

	$t_{1}$	$t_{5}$	$t_{10}$
$d=1$	2.895	3.676	4.200
$d=2$	2.179	2.695	2.989
$d=3$	0.313	0.721	1.115
$d=4$	0.319	0.743	1.151

Table 1: KdV: relative

L^{2}

error (%) at three target frames, averaged over the 200 test trajectories, for symbol degrees

d=1,\dots,4

	$c_{0}$	$c_{1}$	$c_{2}$	$c_{3}$
truth	$0$	$0$	$0$	$-1$
recovered	$-0.061$	$-0.081$	$4.9\times 10^{-5}$	$-0.979$

Table 2: KdV: recovered symbol coefficients at

d=3

against the truth.

Method	$t_{1/2}$	$t_{1}$	$t_{5}$	$t_{10}$
ENNO ( $d=3$ )	0.197 $\pm$ 0.008	0.313 $\pm$ 0.015	0.721 $\pm$ 0.032	1.115 $\pm$ 0.049
TI-DeepONet	2.374 $\pm$ 0.066	1.369 $\pm$ 0.035	2.556 $\pm$ 0.078	3.329 $\pm$ 0.105
E-WSINDy	0.830 $\pm$ 0.061	1.459 $\pm$ 0.079	2.436 $\pm$ 0.124	2.793 $\pm$ 0.143
OSG-FNO	3.264 $\pm$ 0.083	2.045 $\pm$ 0.089	2.008 $\pm$ 0.088	2.112 $\pm$ 0.080
KNO	—	2.049 $\pm$ 0.096	1.824 $\pm$ 0.083	1.912 $\pm$ 0.087
NOMAD	4.800 $\pm$ 0.147	4.815 $\pm$ 0.152	4.804 $\pm$ 0.152	4.832 $\pm$ 0.149

Table 3: KdV: relative

L^{2}

error (%) at four target times, reported as mean

\pm

SE over the 200 test trajectories.

t_{1/2}

is half the smallest training step and is not an observation time; methods that cannot be evaluated below the observation interval are marked “—”.

	$t_{1}$	$t_{5}$	$t_{10}$
$d=1$	2.439	4.011	4.647
$d=2$	0.021	0.074	0.129
$d=3$	0.021	0.074	0.128

Table 4: Advection–diffusion: relative

L^{2}

error (%) at three target frames, averaged over the 25 test trajectories, for symbol degrees

d=1,2,3

	$c_{(0,0)}$	$c_{(1,0)}/c_{(0,1)}$	$c_{(2,0)}=c_{(0,2)}$
truth	$0$	$0/0$	$0.05$
recovered	$0.029$	$0.054/0.061$	$0.0488$

Table 5: Advection–diffusion: recovered symbol coefficients at

d=2

against the truth. The two first-order coefficients are listed as

x/y

Method	$t_{1/2}$	$t_{1}$	$t_{5}$	$t_{10}$
ENNO ( $d=2$ )	0.011 $\pm$ 0.001	0.021 $\pm$ 0.002	0.074 $\pm$ 0.008	0.129 $\pm$ 0.013
E-WSINDy	0.030 $\pm$ 0.003	0.030 $\pm$ 0.003	0.145 $\pm$ 0.016	0.283 $\pm$ 0.030
KNO	—	0.269 $\pm$ 0.037	0.376 $\pm$ 0.049	0.562 $\pm$ 0.072
OSG-FNO	0.262 $\pm$ 0.037	0.524 $\pm$ 0.060	2.617 $\pm$ 0.303	5.223 $\pm$ 0.607
TI-DeepONet	14.597 $\pm$ 2.122	14.551 $\pm$ 2.068	29.970 $\pm$ 3.330	50.831 $\pm$ 5.843
NOMAD	79.330 $\pm$ 5.491	60.124 $\pm$ 4.971	59.634 $\pm$ 4.909	58.969 $\pm$ 4.898

Table 6: Advection–diffusion: relative

L^{2}

error (%) at four target times, reported as mean

\pm

SE over the 25 test trajectories.

t_{1/2}

is half the smallest training step and is not an observation time; methods that cannot be evaluated below the observation interval are marked “—”.

Method	5 minutes	25 minutes	50 minutes
ENNO	1.4 $\pm$ 0.05	5.4 $\pm$ 0.15	9.2 $\pm$ 0.25
KNO	1.8 $\pm$ 0.05	6.7 $\pm$ 0.17	10.3 $\pm$ 0.28
OSG-FNO	1.8 $\pm$ 0.04	8.6 $\pm$ 0.15	13.7 $\pm$ 0.26
NOMAD	5.6 $\pm$ 0.14	8.9 $\pm$ 0.24	13.4 $\pm$ 0.40
TI-DeepONet	6.4 $\pm$ 0.16	11.4 $\pm$ 0.29	19.5 $\pm$ 0.50
E-WSINDy	12.5 $\pm$ 0.30	Failed	Failed

Table 7: SEVIR test MSE (

\times 10^{-3}

) at three forecast lead times, reported as mean

\pm

SE over the 500 test events. Bold marks the best result. “Failed” denotes nonfinite predictions during rollout.

Figure 1: Qualitative SEVIR forecasts for one test event at lead times of 5, 25 and 50 minutes, matching Table 7 . Top row: ground truth; bottom row: ENNO’s autoregressive predictions.

	AUC-PR
Method	$T_{\rm in}=1$	$T_{\rm in}=5$
ENNO	$\mathbf{0.3210}\pm 0.0848$	$\mathbf{0.3122}\pm 0.0835$
KNO	$0.2895\pm 0.0867$	$0.2889\pm 0.0744$
OSG-FNO	$0.2664\pm 0.0811$	$0.3098\pm 0.0887$
TI-DeepONet	$0.1435\pm 0.0373$	Unsupported
NOMAD	$0.1290\pm 0.0394$	Unsupported

Table 8: WildfireSpreadTS test AUC-PR, reported as mean

\pm

sample standard deviation across four held-out years. Bold marks the highest mean. “Unsupported” indicates that the evaluated implementation does not support five-frame inputs.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

	KdV	Advection–diffusion	SEVIR (VIL)	WildfireSpreadTS
Latent channels $M$	1	1	20	4 ( $T_{\mathrm{in}}=1$ ) / 20 ( $T_{\mathrm{in}}=5$ )
History length (frames)	1	1	10	$T_{\mathrm{in}}\in\{1,5\}$
Encoder $E$ / decoder $R$	identity (learnable variants in Table 11 )	identity	pointwise MLPs $\mathbb{R}^{10}\!\to\!\mathbb{R}^{20}$ and $\mathbb{R}^{20}\!\to\!\mathbb{R}$ , 5 hidden layers $\times$ 256	same design, $\mathbb{R}^{T_{\mathrm{in}}}\!\to\!\mathbb{R}^{M}$ and $\mathbb{R}^{M}\!\to\!\mathbb{R}$ , 5 hidden layers $\times$ 256
Cutoff $K$ ( $\|k\|\leq K$ ) a	64	64	64	32
Symbol degree $d$	3 (ablation over $1$ – $4$ , Table 1)	2 (ablation over $1$ – $3$ , Table 4)	6	6
Constant term $c_{0}$	trained	trained	fixed at 0	fixed at 0

Appendix

Table 9: Architecture of ENNO in the four experiments.

	KdV	Advection–diffusion	SEVIR	WildfireSpreadTS
Domain and grid	$[0,10]$ periodic, 128 points	$[0,1]^{2}$ periodic, $128\times 128$	$384\times 384$ km 2 ; VIL on $384^{2}$ pixels, average-pooled to $128^{2}$	daily fire maps at 375 m, padded to $360\times 320$
Source	Fourier pseudospectral in space, DOP853 in time (rtol $10^{-8}$ ); 10,001 frames over $T=1$ , every 10th kept	explicit Euler, $\Delta t=10^{-5}$ , central differences in space; 5,001 frames over $T=0.5$ , every 10th kept	real radar, 49 frames per event	real observations, 2018–2021
Initial conditions	3 sine modes (amplitude $\mathcal{U}(-1,1)/m^{2}$ for mode $m$ ) plus 1 or 2 solitons (amplitude $\mathcal{U}(1.5,3.5)$ )	sums of 1–3 Gaussian bumps (amplitude $\mathcal{U}(0.5,1.5)$ , $\sigma\sim\mathcal{U}(0.05,0.25)$ ) and low-mode Fourier fields	—	—
Observation interval	$10^{-3}$	$10^{-3}$	5 min	1 day
Training step sizes $\tau_{1},\tau_{2}$	$10^{-3},\;2\times 10^{-3}$	$10^{-3},\;2\times 10^{-3}$	5, 10 min	1, 2 days
Targets at $t_{1/2}$	every 5th frame of the same runs	every 5th frame of the same runs	—	—

Appendix

Table 10: Data used in the four experiments.

Variant	$t_{1}$	$t_{5}$	$t_{10}$
$d=1$ , identity	2.898	3.683	4.189
$d=1$ , $M=1$	3.069	2.629	3.276
$d=1$ , $M=4$	1.866	2.317	2.468
$d=1$ , $M=16$	1.877	2.289	2.449
$d=2$ , identity	2.181	2.700	2.994
$d=2$ , $M=1$	1.990	2.454	2.856

Appendix

Table 11: KdV: encoder/decoder ablation. Relative

L^{2}

error (%) at three target frames. “Identity” is the parameter-free configuration reported in the main text; the other rows make them learnable, with

M

latent fields. Best in each block in bold.

Neural Operators

Neural operators learn mappings between functions, enabling fast surrogates for PDEs and other continuous physical systems.

See all 316 papers

Local gradient neural operator
Sep 7, 2026Baiming Zhang, Jinsong Tang, Ying Xu +2PDE Surrogate Modeling PDE Operator Learning
Zhejiang University · Nanjing University of Science and Technology · Hefei University of Technology
Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs
Aug 11, 2026Junfeng ChenPDE Surrogate Modeling PDE Operator Learning
Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, China
Neural Operator Processes for Probabilistic Operator Learning under Partial Observations
Jun 22, 2026Jose Miguel Lara-Rangel, Serge GuillasNeural Processes PDE Operator Learning
University College London London, United Kingdom

Sep 7, 2026J/K move · Enter open · S save

Local gradient neural operator

Baiming Zhang, Jinsong Tang, Ying Xu, Lihua Chen+1

Zhejiang University · Nanjing University of Science and Technology · Hefei University of Technology

PDE Surrogate Modeling PDE Operator Learning PDE Inverse Problems Neural Operators

Open paper AskPDF arXiv

Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.

Abstract

Neural Operators

Explore similar work

Local gradient neural operator

Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs

Neural Operator Processes for Probabilistic Operator Learning under Partial Observations