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Local gradient neural operator

Baiming Zhang, Jinsong Tang, Ying Xu, Lihua Chen, Shiying Xiong

arXiv:2609.07752v1cs.LGmath-phmath.MP

TL;DR

Black-box PDE prediction and source identification are difficult when governing equations are unavailable, while global neural operators may need substantial data. LGNO learns local stencil-like responses for these tasks, with zero-consistent reconstruction and symmetry-based folding where applicable. Across benchmark tasks, it reports lower test error under a unified low-data comparison using compact models.

  • Problem

    Conventional numerical methods require governing equations and coefficients that may be inaccessible in black-box systems, while global neural operators can overfit with limited data and struggle to recover local operator laws.

    Source: §1 · pp. 1–2

  • Method

    LGNO learns local mappings from neighborhood information to temporal or source responses, using symmetry-related samples when the physical setting permits.

    Source: §1 · p. 2

  • Results

    About 74.5% average test-error reduction under the unified low-data comparison, with compact models across the reported benchmarks.

    Source: §4 · p. 14

  • Takeaways & Limitations

    The tested framework spans diffusion, flow, and quantum dynamics, and its core modules provide sample reuse, reduced model size, near-zero drift suppression, and stability under moderate input perturbations.

    Source: §1 · p. 2, §4 · p. 14

    A shared local mapping assumes the grid, boundary treatment, and calibrated operator are compatible with translations.

    Source: §2.1 · p. 4

Abstract

from arXiv · show

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.

1 Introduction

The paper targets black-box PDE prediction and source identification, where conventional solvers require governing equations and global neural operators may struggle with limited data. It proposes a local gradient neural operator and evaluates it across mechanical benchmarks.

1 Introduction

  • Conventional numerical methods require governing equations, coefficients, and discretization details that may be inaccessible in black-box engineering systems.Global neural operators can also overfit with few samples, making it difficult to recover intrinsic local operator laws.Source: §1 · pp. 1–2
  • LGNO formulates temporal prediction and source identification as local mappings from neighborhood information to a time derivative, one-step update, or unknown source term.The two tasks use different physical inputs and targets while sharing a local-operator framing.Source: §1 · p. 2
  • A lightweight micro-network predicts local stencil coefficients, which are contracted with neighboring field values to reconstruct the target response.This gradient-oriented reconstruction provides a discretization interpretation and reduces zero-point drift.Source: §1 · p. 2
  • When physical symmetries permit, LGNO folds equivalent components into serial local samples, reducing input channels and trainable parameters while increasing effective samples.The authors describe this as beneficial in single- or few-sample settings.Source: §1 · p. 2
  • The evaluation covers diffusion, Burgers-type dynamics, Navier–Stokes flow, Gross–Pitaevskii dynamics, and Schrödinger-type evolution without requiring explicit PDE forms or known physical coefficients.Source: §1 · p. 2

2 The proposed LGNO framework

LGNO uses task-specific local mappings and shared stencil-based reconstruction to address temporal evolution prediction and source identification. Its framework relies on local sampling, zero-consistent reconstruction, and symmetry-based sample reuse where the calibrated problem supports those structures.

2.1 Overview of the LGNO framework

  • Temporal prediction takes the primary field and any prescribed source as input and targets the local time derivative, whereas source identification uses the observed field to predict the unknown source.Source: §2.1 · p. 3
  • Each local window is an independent training sample, and sliding-window predictions are assembled into global outputs.This reuses local observations from one or a few fields rather than requiring many full trajectories.Source: §2.1 · pp. 4–5
  • A shared local mapping is justified when the grid, boundary treatment, and calibrated operator are compatible with translations.The framework assembles local responses globally and handles boundary nodes according to each problem’s discretization.Source: §2.1 · pp. 4–5
  • LGNO predicts state-dependent stencil coefficients and contracts them with neighboring field values, separating coefficient learning from target reconstruction.The decomposition preserves local gradient effects and gives the learned operator a discrete-stencil interpretation.Source: §2.1 · p. 5

2.2 Zero-consistent reconstruction and error estimate

  • Bias-free stencil reconstruction enforces zero consistency and suppresses artificial drift near the zero state; the approximation error separates discretization and learned-coefficient errors.Nonzero zero-state responses from coordinates or other conditioning channels can be handled by subtracting the zero-state response first.Source: §2.2 · pp. 5–6

2.3 Symmetry-induced network folding and orbit augmentation

  • Orbit augmentation supplies symmetry-related training pairs, while componentwise folding maps equivalent components to a shared local learning problem.Folding requires the calibrated target to respect the symmetry; the paper excludes symmetry-breaking sources, potentials, anisotropy, or direction-dependent forcing unless represented in the inputs.Source: §2.3 · p. 6

3 Results

LGNO is evaluated across one- to three-dimensional PDE benchmarks for source identification and field prediction, with fixed-seed experiments and test data excluded from training. Results cover diffusion, fluid flow, Burgers dynamics, quantum operators, and robustness to noisy inputs.

3.1 Statement of network architecture and data training

  • The experiments use fixed random seeds, exclude test data from training, and compare model widths to reduce parameter-scaling effects.The benchmark suite covers linear and nonlinear, source-identification and temporal-evolution tasks; the width-controlled comparison focuses on architectural inductive bias.Source: §3.1 · p. 6

3.2 1D examples

  • For one-dimensional diffusion, LGNO-2 achieves 1.21 × 10−2 overall rollout relative L2 error, versus 9.47 × 10−1 for the most accurate baseline, MLPConv-2.LGNO-2 has the lowest error on all six test trajectories, an approximately 98.7% reduction overall.Source: §3.2 · pp. 7–8, Table 3 · p. 7

3.3 2D examples

  • Trained on about 2% of the full trajectory, LGNO-16 preserved the main vortex structures and remained stable until t = 2.0; its best finite rollout relative L2 error was 8.14 × 10−3 across both velocity components, all grid points, and 1000 frames.LOINN-16 had the lowest training MSE, 3.06 × 10−8, and one-step time-derivative relative L2 error, 2.50 × 10−3, but its autoregressive rollout diverged.Source: §3.3 · p. 8
  • LGNO-16 achieves the best finite Navier–Stokes rollout relative L2 error, 8.14 × 10−3, using 1,188 trainable parameters.The baseline MLPConv-16 reaches 1.74 × 10−2; increasing LGNO width to 32 does not substantially improve rollout accuracy.Source: §3.3 · p. 8, Table 4 · p. 9
  • In the Burgers width-8 comparison, LGNO reaches 9.29×10−2 test relative L2 error versus 3.54 × 10−1 for MLPConv, a 73.77% reduction.MLPConv has lower training loss, but LGNO’s Pareto comparison shows better test accuracy despite using 233 rather than 161 parameters.Source: §3.3 · pp. 9–10, Figure 4 · p. 9
  • The Gross–Pitaevskii model’s overall relative L2 error across four unseen potentials and both output components is 11.61%.Errors are below 2% for both components on deformed and disorder potentials, while the bichromatic case is hardest, with 10.19% real-part and 21.57% imaginary-part error.Source: §3.3 · p. 11, Table 5 · p. 11

3.4 3D examples

  • For three-dimensional Schrödinger evolution, the average relative L2 rollout error is 0.3%, with a maximum of 8.58 × 10−3 at the final frame t = 0.5.A central-slice comparison also reports maximum relative error below 5%.Source: §3.4 · p. 12, Figure 6 · p. 13

3.5 Robustness

  • Under tested multiplicative noise levels α ∈ [0, 0.10], LGNO-8 has lower mean relative L2 error than MLPConv-8 at every level.At α = 0.10, errors are 3.166 × 10−1 for LGNO-8 and 3.853 × 10−1 for MLPConv-8; relative improvements range from 17.8% to 73.6%.Source: §3.5 · pp. 12–13

4 Conclusion

LGNO combines gradient-aware stencil reconstruction and symmetry folding for source identification and temporal prediction. Across diverse benchmarks, it reduces test errors with compact models, while noise tests and future-work plans define the reported scope.

4 Conclusion

  • LGNO reduces test error by about 74.5% on average, with reductions of 85.9% for 2D Burgers and 84.2% for 3D Schrödinger, and rollout error by 53.2% versus the best stable Navier–Stokes baseline.The latter comparison is from the Navier–Stokes rollout test.Source: §4 · p. 14
  • The reported models use 23 parameters for 1D diffusion, 105 for 2D Burgers, 370 for 2D Gross–Pitaevskii, and fewer than 1200 for Navier–Stokes and Schrödinger.Source: §4 · p. 14
  • Symmetry folding increases effective samples while controlling model size, and zero-consistent reconstruction removes static offsets and avoids artificial zero-state drift.The error bound separates discretization and fitting errors; Burgers noise tests show stability under moderate input perturbations, while future work targets weakly nonlocal and multi-scale systems with lightweight global context.Source: §4 · p. 14

Data availability

The paper states that its datasets are generated directly from code and formalizes how local differential operators yield discrete stencil rules. The formalization links local inputs to responses and describes conditions for sharing rules across grid points.

Data availability

A Local stencil representation of discretized PDE operators

  • A compact-stencil discretization of a finite-order local differential operator makes each grid response depend on a finite local input window.The local discrete rule is consistent with the continuous operator under the stated smooth-field assumption and discretization order.Source: A Local stencil representation of… · pp. 15–16
  • In a state-dependent affine stencil form, local coupling blocks depend on the local input, while the additive term is independent of contracted field values.When the operator, grid, and discretization are translation equivariant away from boundaries, the same interior local rule can be reused across grid points.Source: A Local stencil representation of… · p. 16
  • Vectorizing the local rules over valid grid nodes yields a block-sparse matrix representation for the discrete operator.Source: A Local stencil representation of… · p. 16

B Proofs of zero-consistent reconstruction and error estimates

The proof calibrates away the local response at zero field, then uses bias-free stencil reconstruction to enforce zero consistency. Its error estimates relate approximation error to discretization and learned-coefficient errors, with rollout bounds requiring explicit stability assumptions.

Zero-consistent reconstruction

  • Pointwise calibration subtracts the zero-state offset before learning the field-dependent operator, including offsets that may vary with conditioning variables.In homogeneous translation-equivariant problems, the offset is spatially constant; coordinate-dependent sources or anisotropy can make it vary.Source: B Proofs of zero-consistent… · pp. 16–17
  • Bias-free stencil reconstruction enforces zero consistency, preventing a translation-invariant background drift when contracted primary-field values are zero.The zero-state output is first removed by calibration and then excluded by the reconstruction structure.Source: B Proofs of zero-consistent… · p. 17

Error estimates

  • Under the stated compact-stencil coefficient assumption, the total error bound links discretization error and learned coefficient error.The derivation combines the assumed discretization estimate with a coefficient-approximation bound and proves the resulting estimate.Source: B Proofs of zero-consistent… · pp. 17–18

Rollout estimate

  • The rollout estimate is conditional on a bounded admissible trajectory, Lipschitz continuity, and a one-step approximation bound.Together with the preceding estimate, it links autoregressive error to compact-stencil discretization error hp and learned coefficient error εθ.Source: B Proofs of zero-consistent… · pp. 18–19

C Derivation of symmetry-induced network folding

Symmetry-induced folding treats transformed inputs and targets as samples of a shared canonical local learning problem. It is valid only when the operator, components, and discretization respect the transformations used.

Symmetry-induced folding

  • For an equivariant calibrated operator, symmetry transformations generate valid input-target pairs, and components in the same orbit can be folded to one canonical local problem.Locality makes the canonical output depend on its corresponding folded window and prescribed conditioning channels.Source: C Derivation of symmetry-induced… · pp. 19–20

Folding conditions

  • Folding requires a canonical map for each component and transformations compatible with the grid and boundary treatment.When transformations do not map nodes exactly, the input representation must include the discretization's interpolation or resampling.Source: C Derivation of symmetry-induced… · p. 19
  • Folding fails to represent one shared operator law when symmetry-breaking inputs are omitted from the equivariant representation.The calibrated operator must be equivariant, and any calibrated or residualized target must still transform under the output representation.Source: C Derivation of symmetry-induced… · p. 20

D Detailed benchmark prediction results of selected models

The appendix reports detailed benchmark prediction results for four selected models across 1D, 2D, and 3D cases. The comparison uses a common low-data protocol rather than fully optimized baseline configurations.

Benchmark results

  • The detailed tables cover MLPConv, DeepONet, LOINN, and LGNO across 1D, 2D, and 3D benchmarks.For each case, they report trainable parameters, final training loss, test relative L2 error, and error ratio against the best-performing model in that benchmark.Source: D Detailed benchmark prediction… · p. 20, Table 6 · p. 21

Experimental protocol

  • All selected models use unified scripts and data processing, without benchmark-specific redesign or extensive tuning of retained baselines.The comparison is intended to reflect behavior under a common low-data protocol, not each architecture's fully optimized limit.Source: D Detailed benchmark prediction… · p. 20

Comparison scope

  • The focused comparison is designed to contrast direct operator mapping with differential local reconstruction when data are limited.LGNO combines a local coefficient generator with differential-stencil reconstruction; the stated conclusions concern observed accuracy, parameter trade-off, and local generalization.Source: D Detailed benchmark prediction… · p. 20

E Additional results for the 2D Navier–Stokes system

Additional 2D Navier–Stokes snapshots compare rollout predictions beyond the concise LGNO-16 and MLPConv-16 comparison. The reported visual contrast concerns how baseline errors develop at later rollout times.

2D Navier–Stokes rollout

  • The additional visual comparisons illustrate LGNO's long-time rollout advantage over the baseline models.The passage presents them as complementing the quantitative comparison in the main text.Source: E Additional results for the 2D… · p. 21
  • MLPConv and DeepONet reproduce the main flow structures at early rollout times, but their predictions grow less accurate as rollout proceeds.The later snapshots show errors in both velocity components.Source: E Additional results for the 2D… · p. 21
  • Later baseline snapshots show blurred local structures, weakened vortical features, and misplaced flow patterns.These are the visual errors reported for the 2D Navier–Stokes example.Source: E Additional results for the 2D… · p. 21

F Additional results for the 2D Burgers equation

Additional 2D Burgers test examples qualitatively compare LGNO-8 with the reference and other models. LGNO-8 remains closer to the reference, especially where spatial variations are sharper.

F Additional results for the 2D Burgers equation

G Dataset examples for the 2D Gross–Pitaevskii equation

The 2D Gross–Pitaevskii training data use an optical lattice potential, while the test set uses four other potential types to assess generalization to unseen potential fields.

G Dataset examples for the 2D Gross–Pitaevskii equation

  • Training data use an optical lattice potential, while test potentials are hexagonal, deformed, bichromatic, and disorder types.The split is designed to evaluate generalization to potential fields not observed during training.Source: G Dataset examples for the 2D… · p. 21, Figure 11 · p. 23
  • Figure 9 presents baseline predictions for the 2D Navier–Stokes system, with MLPConv and DeepONet rows and snapshots at t = 0.8, 1.2, 1.6, and 2.0.Separate panels show horizontal velocity u and vertical velocity v.Source: Figure 9 · p. 22
  • Figure 10 provides additional 2D Burgers test examples comparing the reference solution with LGNO-8 and five other model variants.Source: Figure 10 · p. 22
  • The training-sample illustration shows the potential field, the real and imaginary parts of ˆHψ, and the magnitude |ˆHψ|.Source: Figure 11 · p. 23

H Robustness results

Across all tested multiplicative-noise amplitudes, LGNO-8 has lower mean error than MLPConv-8, with a smaller advantage as noise increases.

H Robustness results

  • The robustness test compares LGNO-8 and MLPConv-8 under multiplicative input noise across five evaluation seeds, reporting case-wise and average relative L2 errors.Table 7 defines positive relative advantage as lower average error for LGNO-8.Source: H Robustness results · p. 23, Table 7 · p. 24
  • Across α ∈[0, 0.10], LGNO-8 has lower mean relative L2 error than MLPConv-8, with an 18% relative advantage at α = 0.10.At this highest tested noise amplitude, MLPConv-8 performs better in two individual cases, but LGNO-8 retains the smaller average error.Source: H Robustness results · p. 23, Table 7 · p. 24
  • For α ≤0.02, LGNO-8 has an average advantage of about 71%–74% over MLPConv-8.The passage characterizes this as the low-noise regime.Source: H Robustness results · p. 23
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