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Green-Routed Neural Operators: Physics Determines Where the Network Reads

Chenhao Si, Ming Yan

arXiv:2610.05337v2cs.LGcs.AI

TL;DR

Many PDEs use transport fields to determine where evolving states are read, while neural operators commonly supply those fields only as input values. GRNO uses the governing equation to route latent reads, and achieves the lowest mean final error on four of five systems in 40-step autoregressive evaluation while remaining competitive on Keller–Segel.

  • Problem

    PDEs use known transport fields to determine read coordinates, but neural operators typically treat them as values at fixed grid indices, leaving this computational role implicit.

    Source: §1 · p. 2

  • Method

    GRNO computes an equation-specified departure map and combines centered and routed latent reads to learn the complete finite-time update.

    Source: §1 · p. 2

  • Results

    Across five systems, GRNO achieves the lowest mean final error on four under 40-step autoregressive evaluation and remains competitive on Keller–Segel; it also has lower mean error than tested controls on all five.

    Source: §1 · p. 2, §6 · p. 9

  • Takeaways & Limitations

    The results support equation-specified latent routing when the route materially affects finite-time prediction.

    Source: §6 · p. 9

    Routing dependence varies by system: Keller–Segel shows little sensitivity to route interventions, and Centered only has the lowest ablation error there.

    Source: §6 · p. 9

Abstract

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We identify a mismatch between the physical role of transport fields in many PDEs and their usual role in neural operators: PDEs use them to select read coordinates, whereas neural operators typically treat them only as input values. We address this mismatch with the Green-Routed Neural Operator (GRNO), which uses the governing equation to determine where latent features are sampled. A parameter-free equation adapter evaluates the diagnostic relation and constructs a departure map whose values are the read coordinates. A multiscale encoder-decoder combines centered and routed reads of latent features to learn the complete finite-time update. Across five two- and three-dimensional PDE systems, GRNO achieves the lowest mean final relative $L^2$ error on four under 40-step autoregressive evaluation and remains competitive on Keller-Segel. Fixed-weight route interventions reveal strong dependence on direction and spatial alignment in four systems, with weak dependence in Keller-Segel. In independently trained ablations, GRNO achieves lower mean errors than variants that supply the transport field only as an input feature, substitute a learned displacement for the equation-specified route, or apply the route with a spatial misalignment, across all five systems. It also substantially outperforms directly advecting the physical state and learning the remaining update, indicating that equation-specified read coordinates provide an effective structural prior for long-horizon PDE forecasting.

1 Introduction

The paper addresses the mismatch between PDEs using known transport fields to select read coordinates and neural operators treating those fields as input values. GRNO makes the equation specify latent read coordinates, and achieves the lowest mean final error on four of five systems in 40-step autoregressive evaluation.

1 Introduction

  • PDEs can specify the coordinates from which an evolving state should be read, but generic endpoint learners leave the known carrier’s role in choosing coordinates implicit.Endpoint supervision constrains the next-state map without specifying the carrier-to-coordinate computation; Flowers makes sampling locations explicit but learns displacements from forecast data [7].Source: §1 · pp. 1–2
  • GRNO uses a parameter-free equation adapter to compute the carrier and departure map, then combines centered and routed latent reads to predict the complete finite-time update.The diagnostic field, carrier, and departure map are recomputed from every predicted state during autoregressive rollout.Source: §1 · p. 2
  • Across five systems, GRNO achieves the lowest mean final error on four under 40-step autoregressive evaluation and remains competitive on Keller–Segel.Fixed-weight route interventions show strong dependence on route direction and alignment in four systems.Source: §1 · p. 2
  • The paper provides an architecture and matched experiments to correct and test the mismatch between a PDE’s coordinate-selecting use of known fields and neural operators’ value-based inputs.Source: §1 · p. 2

2 Related Work

The related work spans learned spatial interactions, multiscale and symmetry-aware representations, learned sampling coordinates, hybrid numerical models, and Green-function methods. GRNO differs by using a known diagnostic carrier to construct read coordinates while learning the complete finite-time update.

2 Related Work

  • Neural operators model nonlocal or spectral interactions with graph, multipole, Fourier, and multiwavelet kernels, while other architectures use multiscale convolutions or attention.Source: §2 · p. 2
  • Related representations encode spatial symmetries, directional structure, or predictive latent features; spatial transformers, deformable convolutions, Geo-FNO, and GINO address sampling transformations or physical-to-latent grids.Source: §2 · p. 2
  • Learned displacements and latent advection appear in Flowers and PARADIS, whereas GRNO constructs read coordinates from the diagnostic carrier.Source: §2 · p. 2
  • Hybrid numerical models learn discretizations or corrections, while GRNO uses physical transport to select latent features and learns the complete finite-time update.Source: §2 · p. 2
  • Prior Green-structure methods learn Green kernels or build Green-inspired operators, whereas GRNO uses a known Green inverse to construct latent read coordinates.Source: §2 · p. 2

3 Motivation: The PDE determines where information is read

The section develops equation-specified routing from a Navier–Stokes example to a general class of PDEs, then analyzes how route errors can affect autoregressive rollouts. It motivates exposing the equation-specified route as a separate latent read while retaining a centered computation.

3.1 A concrete example: diagnostic routing in Navier–Stokes

  • GRNO’s Navier–Stokes example backtracks the transporting velocity to obtain departure coordinates, reads old vorticity there, and leaves viscosity, forcing, drag, and other finite-time effects in the remainder.Uniform rightward transport by one grid spacing makes each target read its left neighbor.Source: §3.1 · p. 3

3.2 Where the mismatch arises: values versus read coordinates

  • Endpoint supervision permits multiple internal route-and-remainder decompositions for identical training pairs, while the governing equation specifies which departure map GRNO uses.Changing the state affects both sampled values and the coordinates used to read them.Source: §3.2 · p. 3

3.3 General form: diagnostically routed PDEs

  • A diagnostically routed PDE has an instantaneous diagnostic relation that determines a carrier, which in turn determines read coordinates; purely local reaction–diffusion equations and nonlocal fields used only as additive sources fall outside this class.GRNO uses the physical transport computation to select latent features while learning the complete finite-time update.Source: Figure 1 · p. 4, §3.3 · p. 5

3.4 How route perturbations accumulate over long rollouts

  • Under the route-perturbation assumptions, next-step error is bounded by propagated state error, coordinate-displacement error weighted by spatial variation, and remainder error.The bound’s propagation factor includes composition, route sensitivity, and remainder sensitivity.Source: §3.4 · p. 5
  • A route defect can propagate through later rollout steps because each predicted state determines the next diagnostic solve and departure map.This motivates preserving the centered computation while exposing the equation-specified route as a separate latent read.Source: §3.4 · p. 5

4 Green-Routed Neural Operator

GRNO uses the equation to set read coordinates and combines centered and routed latent features to learn the complete finite-time update.

4 Green-Routed Neural Operator

  • The equation determines read coordinates, while the neural core learns the complete finite-time update.Source: Figure 1B · p. 5

4.1 Equation-specified routes

  • The normalized neural input concatenates the state, carrier, static context, and normalized time step; ∆tµ is the reference frame spacing and cµ the prescribed context.Source: §4.1 · p. 5

4.2 Multiscale mixing of centered and routed reads

  • At each fixed output-grid point, local features are paired with features interpolated from the departure location and combined across four decoder scales.Source: §4.2 · p. 6

4.3 State increment, rollout, and training

  • The output head predicts the complete state increment, including transport and remaining dynamics; diagnostic equations are recomputed from the next prediction, and training uses a K = 4 rollout.Source: §4.3 · p. 6

5 Experiments

The experiments evaluate GRNO’s long-horizon accuracy and isolate how equation-specified routing contributes across five PDE systems. GRNO leads four systems, and independently trained controls test carrier inputs, route construction, and latent-feature versus physical-state routing.

5 Experiments

  • The five benchmarks span periodic and bounded domains, physical and phase space, and five diagnostic relations; models train through K = 4 steps and undergo free H = 40 rollouts.Baseline comparisons use matched protocols, average over 12 prescribed windows per seed, and report three seeds; independently trained ablations use seed 0.Source: §5.1 · p. 6, §5 · p. 6

5.2 Predictive utility

  • Lower errors extend across substantial portions of four 40-step rollouts, beyond the four supervised steps; GRNO also recovers structures with localized discrepancies in two systems.The visualized systems are Kolmogorov flow and porous convection.Source: §5.2 · pp. 6–7, Figure 2 · p. 8
  • Approximately 46%, 28%, 47%, and 80% lower mean final error than the strongest baseline on Kolmogorov flow, SQG, Vlasov–Poisson, and porous convection, respectively.GRNO remains competitive on Keller–Segel, where Flowers, RieszNO, and GRNO errors are 0.32, 0.34, and 0.37 in the table’s units.Source: §5.2 · p. 6, Table 1 · p. 7

5.3 Does the fitted model use the route?

  • With weights fixed, zero, reverse, and shift route interventions sharply increase final error in four systems, while Keller–Segel changes little.Each intervention increases mean final error by at least 16.7× on Kolmogorov flow, 5.0× on SQG, 41.0× on Vlasov–Poisson, and 13.0× on porous convection.Source: §5.3 · p. 7, Table 2 · p. 7, Figure 2 · p. 8

5.4 Does equation-specified routing improve prediction?

  • GRNO has lower mean errors than independently trained Learned route and Shifted equation-specified route controls on all five systems.The controls replace the equation-specified displacement with a learned displacement or spatially misalign the equation-specified route.Source: §5.4 · pp. 7–8
  • Compared with State advection + residual, GRNO reduces final error by 31.0%, 62.5%, 75.9%, 37.1%, and 64.1% across the five systems, respectively.State advection + residual routes the physical state and learns the remaining update; it has the highest mean error among evaluated variants on every benchmark.Source: §5.4 · p. 8, Table 3 · p. 9
  • Adding equation-specified routing when carrier features are present reduces mean final error by 16.2%, 37.5%, 42.6%, 1.8%, and 46.9% across the five systems, respectively.GRNO leads four systems, while Centered only leads on Keller–Segel; on Kolmogorov flow, improvement over Centered only emerges when carrier features and routing are combined.Source: §5.4 · p. 7, Table 3 · p. 9

6 Discussion

GRNO uses the diagnostic carrier both as an input feature and to route latent reads, with results varying by system. The discussion also records the authors’ AI use and the synthetic-data scope.

6 Discussion

  • GRNO uses the diagnostic carrier both as an input feature and to determine latent read coordinates across flow, phase-space, and three-dimensional transport settings.The equation determines read coordinates while the network learns the complete finite-time update.Source: §6 · p. 9
  • GRNO has lower mean error than Learned route and Shifted equation-specified route on all five systems, and than State advection + residual on every benchmark.Fixed-weight interventions show strong dependence on route direction and alignment in four systems.Source: §6 · p. 9
  • Keller–Segel is the exception: it shows little route sensitivity, and Centered only has the lowest ablation error.In the other four systems, GRNO has the lowest mean ablation error.Source: §6 · p. 9

AI use statement

  • The authors used generative AI tools for text, mathematical and experimental feedback, model-code implementation and debugging, and plotting code.They report conducting the numerical experiments and reviewing AI-assisted code, text, and figures.Source: AI use statement · p. 9

Ethics statement

  • The work uses synthetic numerical PDE data and does not involve human subjects, personal data, or sensitive datasets.Source: Ethics statement · p. 9

A Proofs for the route-sensitivity bounds

The supplement proves route-sensitivity bounds by bounding the one-step error terms and iterating the resulting recursion. It also organizes supplementary material around the paper’s claims.

A Proofs for the route-sensitivity bounds

  • The proof bounds the one-step terms from state error, coordinate change, and remainder difference, then uses repeated substitution to obtain the multi-step bound.For measure-preserving continuum characteristics, change of variables gives B_n = 1.Source: A Proofs for the route-sensitivity… · p. 13

B Supplementary guide

  • The supplement includes proofs, PDE and dataset specifications, adapter and architecture details, training protocols, baseline adaptations, metrics, controls, and visualizations.Source: B Supplementary guide · p. 13
  • Table 4 separates evidence for predictive utility from evidence for the proposed routing mechanism.Source: Table 4 · p. 13

C PDE instantiations and datasets

The supplement specifies PDE-specific diagnostic relations, datasets, adapter computations, backward traces, and routed-interpolation rules. Together these define how equation-derived carriers become latent read coordinates across the evaluated systems.

C.1 Variants of the common template

  • The common template uses advective notation for readability and extends to phase-space states, where the diagnostic solve may determine either carrier component.Table 5 records each task’s instantiation.Source: §C.1 · p. 13

C.2 Equations and coefficients

  • Kolmogorov flow and surface quasi-geostrophy use periodic active-scalar equations with specified coefficients, frame spacing, spectral solvers, and initial-condition procedures.The supplied descriptions include distinct internal-step counts and warm-up procedures.Source: §C.2 · pp. 13–14
  • Keller–Segel, porous convection, and Vlasov–Poisson use different transport and boundary settings, with task-specific coefficients and reference-solver procedures.Keller–Segel uses a screened Poisson relation; the Vlasov–Poisson generator uses spectral Strang translations.Source: §C.2 · p. 14

C.3 Dataset contract

  • Dataset shapes and fixed splits are specified, and each run’s manifest must match the task reference contract for its result to enter the matched table.Manifests log shapes, coefficients, split membership, solver version, and file hashes.Source: Table 6 · p. 15, §C.3 · p. 15

D Equation adapter, traces, and boundaries

D.1 Diagnostic solves

  • Diagnostic solves are task-specific, including Fourier-space Green multipliers, phase-space electric fields, screened Keller–Segel solves, and finite-volume porous-convection pressure solves.The porous-convection adapter derives cell-centered carrier velocities from face velocities.Source: §D.1 · p. 15

D.2 Backward traces

  • Backward traces use a midpoint rule for spatial transport and a symmetric phase-space trace for Vlasov–Poisson.The phase-space read location follows the signs of the characteristic equations when tracing backward.Source: §D.2 · pp. 15–16

D.3 Interpolation and boundary rules

  • Routed reads use bilinear or trilinear interpolation, with scale-adjusted displacements and boundary handling that wraps, clamps, or masks interpolation contributions.The same boundary rule is applied at every latent scale.Source: §D.3 · p. 16, Table 7 · p. 16

E Shared GRNO architecture

GRNO shares a trainable multiscale architecture across PDEs while varying task-specific inputs and deterministic constraints. The supplement also specifies its optimization protocol and composite loss.

E Shared GRNO architecture

  • The shared architecture uses identical parameter names, tensor shapes, stage depths, and channel widths across PDEs within each dimension, while task-specific adapter and metadata vary.Inputs concatenate the state, carrier components, context, and normalized time step; encoder scales produce centered and routed feature tensors.Source: E Shared GRNO architecture · p. 16

W1 LN(DW(u))

  • The increment head projects the finest decoder feature to a normalized state increment, with fixed zero-mean projection for exactly conserved integrals and identity projection for porous convection.The local residual block uses depthwise spatial convolution and channel-wise layer normalization.Source: E Shared GRNO architecture · p. 16, W1 LN(DW(u)) · pp. 16–17

F.1 Optimization and exposure

  • Training defines epochs by data exposure and optimizer updates, and validation selects checkpoints using relative L2 averaged over windows and ten horizons.The matched training and evaluation protocol is summarized in Table 9.Source: §F.1 · p. 17, Table 9 · p. 17

F.2 Loss

  • The loss combines state, gradient, and spectral terms, with physical-field differences and task-specific periodic-axis handling.Gradients pass through four rollout steps, including recomputed diagnostic solves, traces, and interpolations.Source: §F.2 · p. 18

F.3 Free autoregressive evaluation

Evaluation uses prescribed test windows for 40-step autoregressive rollouts, averaging errors within seeds and reporting seed-level statistics. Baselines are matched to a common evaluation protocol, with run provenance recorded.

F.3 Free autoregressive evaluation

  • Errors are averaged over 12 prescribed test windows within each seed; Tables 1 and 2 report means and sample standard deviations across three seed means, while Table 3 reports seed 0.The 2D systems start at frame 0; porous convection and Keller–Segel use specified later start indices.Source: §F.3 · p. 18

G Baseline provenance and matched adaptations

H Metrics and result records

The section specifies rollout metrics and intervention procedures, then describes independently trained controls and a visualization protocol. It records task-specific metrics in addition to relative L2 error.

H Metrics and result records

  • Relative L2 error is evaluated in physical units at each rollout step, with errors computed per trajectory and step before window and seed aggregation.Tables 1 and 2 report means and sample standard deviations across three seed means; Table 3 reports seed 0.Source: H Metrics and result records · p. 19
  • The evaluator also records task-specific physical metrics; spectral error is the square root of the training spectral ratio, while relative H1 includes discrete derivative energies.Spatial correlation is computed after subtracting each field’s spatial mean, and persistence uses b_qh = q0.Source: Table 11 · p. 19, H Metrics and result records · p. 19

I.1 What each intervention changes

  • At every rollout step, the carrier and departure displacement are recomputed from the predicted state, and the chosen intervention is applied before multiscale pooling and feature sampling.The Shift intervention uses fixed dimension-specific spatial offsets.Source: §I.1 · p. 19
  • The intervention errors are reported in Table 2, with corresponding error ratios and rollout profiles shown in Figure 2c–d.Source: §I.1 · p. 19

I.2 Independently trained controls

  • Factorial controls retain identical parameter counts by zeroing carrier inputs or latent displacement; a shifted equation-specified route applies the same deterministic shift in training and testing.The learned-route control predicts displacement with a jointly optimized head; State advection + residual first advects the state and trains a centered network on the complete increment.Source: §I.2 · pp. 19–20

I.3 Visualization sample and transect

  • Figure 2e selects the window with the largest final-error reduction from Carrier feature only to GRNO, then selects a transect maximizing mean absolute-error reduction across 40 steps.The models share the selected window and line; concentration maps use a common color scale and reference contour.Source: §I.3 · p. 20

J.1 Additional qualitative forecasting results

The appendix presents qualitative forecasts across five PDE systems and visualizes route interventions using fixed fitted checkpoints. The figures specify field, prediction, and error layouts but do not state comparative outcomes.

J.1 Additional qualitative forecasting results

  • Figures 3–7 compare reference fields with pointwise absolute errors for GRNO and the strongest Table 1 baseline at ten horizons spanning the 40-step rollout.The systems shown are Kolmogorov flow, dissipative SQG, Vlasov–Poisson, Keller–Segel, and porous convection.Source: §J.1 · p. 20, Figure 3 · p. 20, Figure 4 · p. 21

J.2 Qualitative effects of route interventions

  • For each system, route-intervention figures show a reference and predictions above corresponding pointwise absolute errors, holding the fitted GRNO checkpoint fixed and changing only the departure route.The displayed interventions are correct, zero, reversed, and shifted routes.Source: §J.2 · p. 23, Figure 8 · p. 23, Figure 9 · p. 23
  • The route-intervention figures cover Kolmogorov flow, dissipative SQG, Vlasov–Poisson, Keller–Segel, and porous convection at h = 40.Keller–Segel and porous-convection visualizations use a fixed slice.Source: Figure 8 · p. 23, Figure 9 · p. 23, Figure 10 · p. 24
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