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Physics-Integrated Operator Learning via Gaussian Splatting Representations

Jihao Zhang, Junyi Guo, Jian-Xun Wang

arXiv:2608.24049v1cs.LG

TL;DR

Purely data-driven neural operators can degrade during long-horizon prediction, while physics-informed approaches face optimization difficulties or PDE-specific design constraints. This paper introduces FFGS as a continuous field representation that provides access to governing operators and reports improved long-horizon accuracy and spectral fidelity, including when governing information is partial.

  • Problem

    Purely data-driven formulations progressively degrade long-horizon accuracy, while PDE-specific designs motivate introducing governing knowledge through the field.

  • Method

    The framework employs FFGS as a continuous representation of physical fields, with analytic Gaussian representations providing direct access to spatial derivatives for available governing terms.

  • Results

    Representation-level physics integration consistently improves long-horizon rollout accuracy and spectral fidelity over purely data-driven baselines, retaining substantial gains with partial governing information.

  • Takeaways & Limitations

    Continuous field representations provide a practical interface for incorporating governing knowledge into neural-operator surrogates, including under incomplete physics.

  • Takeaways & Limitations

    Physics-informed training can involve delicate objective weighting because data and physics objectives have disparate scales and gradient dynamics.

Abstract

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Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors during long-horizon autoregressive prediction and may fail to exploit available governing-equation structure. Existing approaches incorporate physics primarily through residual-based training objectives or PDE-specific architectural constraints, which can introduce optimization difficulties or limit architectural generality. In this work, we introduce a representation-level approach to physics integration in which a feed-forward Gaussian splatting (FFGS) representation serves as a continuous interface between discretized solution fields and governing operators. The FFGS representation reconstructs the state as a continuous Gaussian field with closed-form spatial derivatives, allowing available physical PDE operators to be integrated directly within the learned evolution map without introducing a physics-residual loss. We evaluate the framework across two- and three-dimensional PDE systems, including advection, diffusion, nonlinear self-advection, and reaction dynamics. Over long-horizon autoregressive rollouts, the proposed framework reduces relative $\ell_2$ error by $1.5\times$--$2.2\times$ compared with the strongest purely data-driven baseline across the benchmark suite, while consistently improving spectral fidelity. The framework also remains effective when the governing equations are partially known, demonstrating robustness to incomplete physics. These results demonstrate that continuous field representations can provide a practical interface for incorporating known physical structure into generic neural-operator surrogates.

1. Introduction

The paper introduces FFGS as a continuous field interface that embeds available governing terms into a generic neural-operator evolution map, improving long-horizon rollout accuracy and spectral fidelity across PDE systems.

  • Motivation: Standard neural operators infer dynamics from finite trajectory data, so repeated autoregressive errors can degrade long-horizon predictions, especially outside the training distribution.The introduction identifies error accumulation during repeated operator evaluations and amplified degradation under distribution shift.
  • Limitations of existing approaches: Residual-based physics integration can suffer from delicate objective weighting and costly derivative evaluation that limits scalability.The cited limitations include disparate data and physics scales, repeated spatial and temporal derivatives, and increasing automatic-differentiation costs with collocation points and derivative order.
  • Motivation: PDE-specific physics-aware architectures can improve learned dynamics but may require redesign when the governing equation changes.The paper motivates a representation-level alternative to preserve architectural generality across governing systems.
  • Proposed framework: The proposed framework uses FFGS to reconstruct discretized fields continuously and provide analytic spatial derivatives for direct governing-term evaluation within a generic neural operator.The continuous representation acts as an interface between the discretized state and governing operators while retaining a generic neural-operator architecture.
  • Evaluation scope: The framework targets fully or partially known physics and is evaluated on two- and three-dimensional systems involving advection, diffusion, nonlinear self-advection, and reaction dynamics.In the partial-physics setting, the governing form is known while its coefficients are identified from trajectory data.
  • Results: Across the evaluated settings, representation-level physics integration improves long-horizon rollout accuracy and spectral fidelity over purely data-driven baselines.Ablations indicate complementary roles for the embedded-physics and neural-operator components, supporting FFGS as a practical interface for known governing structure.

2. Methodology

The method integrates available PDE structure at the representation level by using FFGS to provide continuous fields and closed-form spatial derivatives within a learned evolution map. A neural operator is combined with an embedded-physics component and trained through autoregressive rollouts without an auxiliary physics-residual loss.

  • Physics-integrated operator: FFGS reconstructs the current solution as a continuous Gaussian field whose closed-form spatial derivatives supply quantities required by the governing operator.The representation provides a continuous differential interface while retaining the grid-resolved state for state-dependent algebraic terms.
  • Physics-integrated operator: The embedded-physics operator integrates prescribed governing terms over Δt and is combined with a neural operator to approximate the finite-time solution operator.The governing PDE defines the finite-time solution operator, while the learned composite map advances the state autoregressively.
  • Differentiable field representation: State-dependent algebraic quantities are evaluated from the current grid-resolved state, while filtered FFGS derivatives provide the spatial differential quantities.This division lets FFGS act as a continuous differential interface without replacing the current state in the governing operator.
  • Differentiable field representation: The encoder predicts anisotropic Gaussian primitives on a spatial anchor lattice, renders local neighborhoods, and reconstructs continuous field quantities.Local rendering avoids evaluating all primitives at every query point while exploiting Gaussian spatial decay.
  • Differentiable field representation: Analytic Gaussian derivatives are evaluated directly without finite-difference stencils or automatic differentiation with respect to spatial coordinates.A low-pass spectral filter is applied to the differential quantities before they enter the governing operator to suppress high-frequency reconstruction errors.
  • Rollout training: The frozen FFGS representation is used during multi-step autoregressive rollout training, while neural-operator parameters are optimized through the rollout objective.The embedded-physics component enters the rollout itself rather than an auxiliary physics-residual term in the loss; stop-gradient avoids backpropagation through repeated evaluations and integrations.
  • Partially known physics: Partially known physics is handled by identifying unavailable coefficients from trajectory data before using them in the same physics-integrated surrogate.The considered governing operators have coefficient dependence that is linear, with the prescribed governing-term forms retained.

3. Experimental setup

The experiments evaluate the surrogate on fixed-coefficient and coefficient-unknown PDE regimes spanning transport, diffusion, nonlinear self-advection, reaction-driven interfaces, and three-dimensional dynamics. Models are trained and tested under shared benchmark protocols using short training windows and long autoregressive rollouts.

  • Benchmark suite: The benchmark suite covers pure transport, transport–diffusion, nonlinear self-advection, reaction-driven moving interfaces, and three-dimensional transport–diffusion.The fixed-coefficient regime uses five benchmarks under a common protocol.
  • Benchmark suite: The coefficient-unknown regime withholds governing coefficients, which are identified from trajectory data before rollout training.The identified coefficients are then fixed for the corresponding trajectory and supplied to the neural operator through coefficient conditioning.
  • Data generation: Each benchmark contains 256 training trajectories and 30 held-out test trajectories on a unit periodic domain with truncated-Fourier initial conditions.The fixed-coefficient problems are nondimensional, with initial-condition modes restricted to |k_i| ≤ 5 along each spatial axis.
  • Evaluation protocol: Models train on the first 50 steps and are evaluated with a 200-step autoregressive rollout under held-out initial conditions.This protocol tests generalization to unseen initial conditions and extrapolation beyond the temporal training window.
  • Training protocol: The proposed surrogate uses a two-stage training protocol in which the FFGS encoder is trained on reconstruction, frozen, and then used during rollout training.Only neural-operator parameters are optimized in Stage 2 using the composite rollout objective, with rollout length H = 5.

Appendix C.

Evaluation uses relative field error and spectral fidelity measures aggregated across long rollouts and held-out trajectories. Lower relative ℓ2 and spectral errors indicate better performance, while PSNR is reported as a complementary metric.

  • Field-level metrics: Relative ℓ2 error at lead time k is the primary field-level metric.Rollout curves average rL2(k) at each lead time over the 30 held-out test trajectories.
  • Aggregation: Summary-table errors are averaged over each trajectory’s 200-step rollout and reported as mean ± one standard deviation across 30 trajectories.The same trajectory-level aggregation is used for the reported rollout errors.
  • Spectral metrics: Spectral fidelity is assessed by computing the radial energy spectrum for each rollout frame and averaging it across frames and held-out trajectories.The radial spectrum is constructed from Fourier coefficients indexed by integer wavenumber.
  • Spectral metrics: For advection, advection–diffusion, and Burgers, spectral evaluation excludes the κ = 0 mean mode because the reference mean is zero and conserved.The advection–Allen–Cahn benchmark retains κ = 0 because its order-parameter mean evolves and is not conserved.
  • Metric interpretation: Lower values indicate better performance for both relative ℓ2 error and spectral error.PSNR is additionally reported using the global dynamic range of the held-out reference data.

4. Results

Across five PDE benchmarks, the proposed surrogate achieves the best reported test-rollout performance, with lower long-horizon error and stronger spectral fidelity than purely data-driven baselines. These gains persist for unseen trajectories, three-dimensional field structure, and partially known physics.

  • The proposed surrogate achieves the best performance on all five benchmarks and all reported metrics.
  • Relative ℓ2 error is reduced by 1.5×–2.2× versus the strongest purely data-driven baseline, with PSNR gains of 3.3–6.6 dB.The strongest baseline varies by benchmark: FNO for 2D linear transport, ResNet for nonlinear cases, and UNet for 3D advection–diffusion.
  • Comparable trainable dynamics-map budgets indicate that the improvement is associated with the FFGS-based embedded-physics component rather than increased learned-model capacity.
  • The proposed surrogate maintains the lowest relative ℓ2 error throughout Burgers rollouts, including extrapolation beyond training and held-out initial conditions.The performance gap becomes increasingly pronounced beyond the training horizon, indicating slower autoregressive error accumulation.
  • In three-dimensional advection–diffusion, the proposed method preserves signed structures, topology, spatial distribution, and characteristic length scales more faithfully than the baselines.At t = 200, all surrogates accumulate temporal error, but the proposed method retains principal positive and negative structures with less severe distortion.
  • The proposed surrogate achieves the smallest spectral error across the resolved 2D Burgers spectrum and remains closest to the reference energy distribution.Data-driven baselines overestimate energy at the first nonzero wavenumber, whereas the proposed surrogate remains nearly indistinguishable from the reference there.
  • Under partially known physics, the proposed surrogate maintains slower rollout error growth and follows the reference spectrum despite estimated physical coefficients.Baselines accumulate spurious high-wavenumber energy, whereas the proposed surrogate follows the reference spectrum down to the numerical floor.
  • The accuracy improvements entail an accuracy–cost trade-off in per-step inference time.

5. Discussion

The composite surrogate benefits from combining embedded governing physics with a neural-operator component and spectral filtering. These gains persist across fixed- and partial-physics settings, while representation capacity, equation-family specialization, and manual cutoff selection remain limitations.

  • Ablation findings: Removing the embedded-physics component increases rollout error across all five fixed-coefficient benchmarks.
  • Ablation findings: Removing the low-pass filter causes a growing Burgers error gap because spatial differentiation amplifies high-wavenumber reconstruction errors.
  • Ablation findings: The composite surrogate achieves lower rollout error and better spectral fidelity than either the embedded-physics or neural-operator component alone.
  • Ablation findings: Both embedded physics and spectral filtering contribute to improved rollout behavior across prescribed-coefficient and coefficient-unknown settings.
  • Limitations and future directions: The method is limited by fixed Gaussian capacity, specialization to a solution family, and manually selected spectral cutoffs.
  • Overall results: Across five two- and three-dimensional PDE benchmarks, the framework improves long-horizon autoregressive accuracy and spectral fidelity beyond comparable purely data-driven neural steppers.
  • Overall results: The approach remains effective with partially known equations, retaining most accuracy and spectral-fidelity gains observed when coefficients are prescribed.

Appendix A.4. Periodic and local rendering

The implementation enforces periodicity and accelerates Gaussian rendering by restricting evaluation to a fixed local neighborhood. Local rendering closely matches dense evaluation while providing a substantial speedup.

  • Periodic rendering: Periodic boundary handling is applied consistently during representation training and embedded-physics evaluation.
  • Local rendering: The renderer evaluates each query using a fixed 9^d local neighborhood with Wloc = 4.
  • Local rendering: 13.9–14.6× rendering speedup is achieved while closely reproducing dense evaluation for reconstructed fields and spatial derivatives.

Appendix B. Filtering and time integration

The embedded-physics component computes governing terms from analytically differentiated FFGS fields, filters unreliable spectral content, and advances the state with RK4. Training freezes this component while optimizing the neural operator separately.

  • Filtering and integration: The frozen FFGS representation supplies reconstructed quantities and closed-form spatial derivatives to the embedded-physics operator.
  • Filtering and integration: The low-pass filter suppresses high-wavenumber derivative errors, limits bandwidth entering the explicit integrator, and removes out-of-band nonlinear modes.
  • Filtering and integration: The available governing operator is advanced with classical fourth-order Runge–Kutta integration, filtering intermediate derivatives and nonlinear right-hand sides as specified.
  • Filtering and integration: Spectral filtering is confined to the embedded-physics component, while the neural-operator component acts directly on the current grid state.
  • Two-stage optimization: Stage 1 trains and then freezes the FFGS encoder; Stage 2 optimizes only neural-operator parameters using a five-step rollout loss.

Appendix D.1. Coefficient-unknown dataset

The coefficient-unknown benchmark identifies trajectory-specific advection and diffusion coefficients from a seven-frame calibration window, then supplies those estimates to both surrogate branches during rollout.

  • Dataset and protocol: Training and test trajectories sample constant per-trajectory coefficients independently from the same distributions, with 50-step training and 200-step evaluation rollouts.
  • Coefficient identification: Each trajectory’s coefficients are identified before Stage 2 using FFGS-derived spatial derivatives and a least-squares regression system.
  • Coefficient identification: Identification uses seven consecutive observed frames and a sixth-order centered finite-difference approximation for the temporal derivative.
  • Coefficient identification: The recovered estimate is λ̂ = (v̂, D̂), with directional advection estimates combined under the isotropic-velocity assumption.
  • Coefficient use: The same identified coefficients are supplied to both the embedded-physics operator and the FiLM-conditioned neural operator; ground-truth values are withheld.
  • Evaluation protocol: Held-out trajectories undergo the same seven-frame calibration, with recovered coefficients fixed throughout prediction and no later reference states used.
  • Evaluation protocol: Long-horizon accuracy is evaluated over 200-step autoregressive rollouts across two-dimensional advection, advection–diffusion, advection–Allen–Cahn, and three-dimensional advection–diffusion.

Appendix E.1. Additional long-horizon rollout results

Additional rollout comparisons show that the proposed surrogate preserves reference structures more accurately than the baselines over long horizons, with advantages across transport, diffusion, reaction, and nonlinear self-advection dynamics.

  • Long-horizon rollout accuracy: The error gap between the proposed surrogate and purely data-driven baselines increases progressively with lead time in two-dimensional advection and advection–diffusion.Three-dimensional advection–diffusion shows the same pattern, with the proposed surrogate exhibiting the slowest long-horizon error growth.
  • Long-horizon rollout accuracy: The proposed surrogate achieves the lowest error over most rollout horizons in the remaining fixed-coefficient benchmarks.On the advection–Allen–Cahn benchmark, it preserves reference interface geometry and large-scale phase organization throughout the rollout.
  • Long-horizon rollout accuracy: The proposed surrogate most closely matches the reference advection–diffusion field over the full rollout, including dominant structures’ locations, signs, and characteristic scales.The baselines accumulate increasing spatial, amplitude, morphological, or local discrepancies at later lead times.
  • Long-horizon rollout accuracy: The proposed surrogate most faithfully tracks dominant Burgers vorticity structures through t = 200, while baselines accumulate phase, amplitude, and morphological errors.It preserves the structures’ location, orientation, and width more accurately than ResNet, U-Net, and FNO.
  • Overall interpretation: Snapshot comparisons indicate that lower aggregate rollout errors correspond to improved preservation of physically relevant spatial structures, especially beyond the training horizon.The advantage appears across transport–diffusion, reaction-interface, and nonlinear self-advection dynamics.
  • Spectral fidelity: The proposed surrogate provides the closest agreement with the reference time-averaged spectrum across all four remaining fixed-coefficient benchmarks.It maintains substantially lower high-wavenumber spectral distortion than purely data-driven baselines throughout the 200-step autoregressive rollouts.

Appendix E.3. Model-capacity control on the Burgers benchmark

Capacity control on Burgers shows that increasing baseline capacity improves rollout accuracy, but the proposed surrogate retains advantages against matched-budget baselines and enlarged U-Net and ResNet models while achieving the lowest spectral error.

  • Rollout accuracy: The proposed surrogate remains more accurate than matched-budget baselines and enlarged U-Net and ResNet models.Its comparison is made against baselines with tenfold trainable dynamics-map budgets, while the surrogate keeps its main-experiment configuration.
  • Rollout accuracy: Increasing purely data-driven baseline capacity consistently improves long-horizon rollout accuracy.The enlarged FNO performs best among purely data-driven models and slightly outperforms the proposed surrogate over part of the rollout horizon.
  • Spectral fidelity: The proposed surrogate achieves the lowest time-averaged spectral error among all compared models at the matched FNO dynamics-map budget.Increasing baseline capacity reduces but does not eliminate spectral discrepancy.
  • Spectral fidelity: The spectral-fidelity advantage cannot be explained solely by increased neural-model capacity.This conclusion follows from the proposed surrogate’s lower spectral error despite the matched-budget comparison and enlarged baselines.

Appendix F. Representation diagnostics

Representation diagnostics evaluate whether the frozen FFGS field is accurate enough for physics evaluation by measuring field, gradient, and Laplacian reconstruction errors on periodic test problems.

  • Diagnostic setup: The diagnostics compare FFGS field and derivative reconstructions with reference spatial derivatives computed by pseudo-spectral differentiation.Table F.6 reports relative errors for the field, gradient, and Laplacian across benchmarks.
  • Derivative sensitivity: Derivative errors increase under spatial differentiation and are largest for the Laplacian.Higher-order derivatives are therefore more sensitive to reconstruction error than the reconstructed field.
  • Derivative sensitivity: Spectral filtering is applied before differential quantities are used in physics evaluation.The filtering is motivated by the increased sensitivity of higher-order derivatives to reconstruction error.
  • Reconstruction accuracy: The frozen FFGS representation maintains low field-level reconstruction errors across all benchmarks.The evaluation tests the representation before using differential quantities to evaluate available governing terms.
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