cs.LGAug 25, 2026

Physics-Integrated Operator Learning via Gaussian Splatting Representations

Authors: Jihao Zhang, Junyi Guo, Jian-Xun Wang

Organizations: Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA

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Abstract

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 ℓ2\ell_2 error by 1.5×1.5\times--2.2×2.2\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.

Figures & tables

Figure 1: Method overview. (a) The FFGS encoder maps a normalized grid field to anisotropic Gaussian primitives, which are rendered locally into a continuous field. The encoder is trained in Stage 1 by field reconstruction and then frozen. (b) The physics-integrated neural operator GΔtθ\mathcal{G}^{\theta}_{\Delta t} adds the embedded-physics component ΦΔtFFGS\Phi^{\mathrm{FFGS}}_{\Delta t} to the neural-operator component Nθ\mathcal{N}_{\theta} , with both acting on the same state. ΦΔtFFGS\Phi^{\mathrm{FFGS}}_{\Delta t} evaluates the available governing terms with the filtered FFGS quantities and advances the state over Δt\Delta t . Each prediction is fed back as the next input, starting from upred(t0)=u(t0)\mathbf{u}^{\mathrm{pred}}(t_{0})=\mathbf{u}(t_{0}) . In Stage 2, both components are evaluated during the multi-step rollout, but only Nθ\mathcal{N}_{\theta} is trained, and no gradients propagate through ΦΔtFFGS\Phi^{\mathrm{FFGS}}_{\Delta t} .
BenchmarkMethod#ParamsPSNR ↑\uparrow (dB)rℓ2↓\mathrm{r}\ell_{2}\downarrowSpec. err ↓\downarrow
Advection (2D)FNO57.8k40.04±1.6840.04\pm 1.680.089±0.0110.089\pm 0.0110.02140.0214
U-Net55.7k36.20±1.1436.20\pm 1.140.129±0.0250.129\pm 0.0250.04030.0403
ResNet61.2k29.42±0.6629.42\pm 0.660.394±0.0280.394\pm 0.0280.28980.2898
Ours57.8k44.75±1.92\mathbf{44.75\pm 1.92}0.052±0.014\mathbf{0.052\pm 0.014}0.0031\mathbf{0.0031}
Adv–Diffusion (2D)FNO57.8k41.41±1.7541.41\pm 1.750.084±0.0110.084\pm 0.0110.02760.0276
U-Net55.7k36.37±1.7036.37\pm 1.700.146±0.0370.146\pm 0.0370.08940.0894
Table 2: Aggregate test-rollout performance on the five full-physics PDE benchmarks.
Figure 2: Error evolution over the rollout on the Burgers benchmark. Relative ℓ2\ell_{2} error is plotted on a logarithmic scale for the proposed surrogate and three purely data-driven baselines. Panel (a) shows rollouts on the training trajectories, and panel (b) shows rollouts on held-out trajectories with unseen initial conditions. The shaded region in panel (a) indicates extrapolation beyond the 50-step training horizon ( t>50t>50 ).
Figure 3: Three-dimensional advection–diffusion rollout for a representative held-out trajectory at large Péclet number. Positive and negative scalar structures are visualized using the isosurfaces u=+0.4u=+0.4 and u=−0.4u=-0.4 , respectively. Columns correspond to the ground truth, the proposed model, FNO, U-Net, and ResNet. Rows show the initial condition and the autoregressive predictions at t=50t=50 , 100100 , and 200200 . All methods are initialized from the same ground-truth field at t=0t=0 , and the viewing direction and isosurface levels are fixed across all panels.
Figure 4: Spectral fidelity on the 2D Burgers test set ( ν=10−4\nu=10^{-4} ; 30 held-out trajectories ×\times 200 rollout steps). (a) Time-averaged radial energy spectrum E‾(κ)\overline{E}(\kappa) for the reference solution and each model. (b) Ratio E‾(κ)/E‾GT(κ)\overline{E}(\kappa)/\overline{E}_{\mathrm{GT}}(\kappa) over the energy-containing range. (c) Relative error of the time-averaged spectrum, ∥E‾−E‾GT∥2/∥E‾GT∥2\|\overline{E}-\overline{E}_{\mathrm{GT}}\|_{2}/\|\overline{E}_{\mathrm{GT}}\|_{2} , evaluated over κ≥1\kappa\geq 1 (Section 3.3 ).
Figure 5: Partial-physics regime on the advection–diffusion benchmark (30 held-out trajectories ×\times 194 rollout steps). (a) rℓ2\mathrm{r}\ell_{2} error of the proposed model and the baselines at matched and tenfold ( 10×10\times ) trainable dynamics-map budgets. (b) Time-averaged energy spectra. The dotted line marks the band edge of the initial conditions.
Figure 6: Per-step inference cost measured as wall-clock time per autoregressive update. Values are the minimum over five timed rollouts after warm-up and are shown on a logarithmic scale.
Figure 7: Ablation in the full-physics regime. (a) Mean rℓ2\mathrm{r}\ell_{2} error over the 200-step rollout with and without the embedded-physics component. Without this component, the model reduces to the FNO baseline, and the values coincide with those in Table 2 . (b) rℓ2\mathrm{r}\ell_{2} error over the rollout on the Burgers benchmark with and without the low-pass filter. Both panels use a logarithmic scale.
Figure 8: Component ablation in the partial-physics regime. All variants use the same identified coefficients. FFGS-only is the embedded-physics component alone. FiLM-FNO is the neural-operator component trained alone, with the coefficients supplied through FiLM conditioning. (a) rℓ2\mathrm{r}\ell_{2} error over the rollout. (b) Time-averaged energy spectra. The dotted line marks the band edge of the initial conditions. (c) Mean rℓ2\mathrm{r}\ell_{2} error over the 200-step rollout.
Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

 Relative ℓ2\ell_{2} discrepancy 
Benchmarku^\hat{u}∂xu^\partial_{x}\hat{u}∂yu^\partial_{y}\hat{u}∇2u^\nabla^{2}\hat{u}Speedup
2D Adv1.8×10−61.8\times 10^{-6}5.7×10−65.7\times 10^{-6}1.0×10−51.0\times 10^{-5}2.8×10−52.8\times 10^{-5}13.9×13.9\times
2D Adv–Diff1.9×10−61.9\times 10^{-6}5.3×10−65.3\times 10^{-6}1.2×10−51.2\times 10^{-5}3.2×10−53.2\times 10^{-5}14.3×14.3\times
2D Burgers7.1×10−87.1\times 10^{-8}1.1×10−71.1\times 10^{-7}1.6×10−71.6\times 10^{-7}5.8×10−75.8\times 10^{-7}13.9×13.9\times
2D Adv–Allen–Cahn7.5×10−87.5\times 10^{-8}1.1×10−71.1\times 10^{-7}2.3×10−72.3\times 10^{-7}8.7×10−78.7\times 10^{-7}14.6×14.6\times
Appendix
Table 3: Local rendering with Wloc=4W_{\mathrm{loc}}=4 compared with dense summation over all primitives. The table reports the relative ℓ2\ell_{2} discrepancies of the reconstructed field and its derivatives and the speedup of the local rendering.
Benchmarkeue_{u}e∇ue_{\nabla u}e∇2ue_{\nabla^{2}u}
2D Adv1.41×10−31.41\times 10^{-3}2.57×10−32.57\times 10^{-3}6.51×10−36.51\times 10^{-3}
2D Adv–Diff1.27×10−31.27\times 10^{-3}2.48×10−32.48\times 10^{-3}6.18×10−36.18\times 10^{-3}
2D Burgers1.51×10−31.51\times 10^{-3}5.13×10−35.13\times 10^{-3}2.62×10−22.62\times 10^{-2}
2D Adv–Allen–Cahn1.53×10−31.53\times 10^{-3}1.17×10−21.17\times 10^{-2}5.38×10−25.38\times 10^{-2}
3D Adv–Diff5.87×10−35.87\times 10^{-3}8.15×10−38.15\times 10^{-3}1.34×10−21.34\times 10^{-2}
Appendix
Table 4: Relative ℓ2\ell_{2} errors of the frozen FFGS reconstruction and its closed-form derivatives, measured against pseudo-spectral references.
BenchmarkN/2N/2Δt\Delta tκc\kappa_{c}Filter
2D Adv801.06Smooth, η=2\eta=2
2D Adv–Diff801.06Smooth, η=2\eta=2
2D Burgers801.010Sharp
2D Adv–Allen–Cahn800.114Sharp
3D Adv–Diff321.08Smooth, η=2\eta=2
2D Adv–Diff (partial physics)801.07Sharp
Appendix
Table 5: Filter settings of the embedded-physics component. N/2N/2 is the largest Fourier index along each axis, and Δt\Delta t is the integration step, which equals the frame spacing.
 Stage 1: FFGS encoderStage 2: rollout training
OptimizerAdamWAdam
Learning-rate scheduleCosine decayLinear warmup, cosine decay
Peak learning rate1.5×10−31.5\times 10^{-3}1.0×10−31.0\times 10^{-3}
Final learning rate1.0×10−61.0\times 10^{-6}00
Training length80 epochs10,000 steps
Batch size3220
Appendix
Table 6: Optimization settings of the two training stages. For the 3D benchmark, Stage 1 uses 40 epochs and batch size 8, and Stage 2 uses a global batch size of 12. For the partial-physics benchmark, Stage 1 uses 100 epochs and batch size 64.
Figure 9: rℓ2\mathrm{r}\ell_{2} error over the 200-step rollout on the remaining full-physics benchmarks, averaged over the 30 held-out trajectories and shown on a logarithmic scale. (a) 2D advection, (b) 2D advection–diffusion, (c) 2D advection–Allen–Cahn, and (d) 3D advection–diffusion. All models are trained on the first 50 steps.
Figure 10: Representative rollout on 2D advection–Allen–Cahn. Columns show the initial condition and the predictions at t=33t=33 , 6666 , 100100 , 133133 , 166166 , and 199199 . Rows show the reference solution, ResNet, FNO, U-Net, and the proposed model. The order parameter uu uses a common color scale across all methods and lead times.
Figure 11: Representative rollout on 2D advection–diffusion. Columns show the initial condition and the predictions at t=33t=33 , 6666 , 100100 , 133133 , 166166 , and 199199 . Rows show the reference solution, ResNet, FNO, U-Net, and the proposed model. The scalar field uu uses a common color scale across all methods and lead times.
Figure 12: Representative rollout on 2D Burgers. Columns show the initial condition and the predictions at t=33t=33 , 6666 , 100100 , 133133 , 166166 , and 199199 . Rows show the reference solution, ResNet, FNO, U-Net, and the proposed model. The displayed quantity is the vorticity normalized by the instantaneous reference standard deviation, ω/σω(t)\omega/\sigma_{\omega}(t) , and the values of σω(t)\sigma_{\omega}(t) are given below each column.
Figure 13: Time-averaged radial energy spectra on the remaining full-physics benchmarks, averaged over the 30 held-out trajectories and 200 rollout steps. (a) 2D advection–Allen–Cahn, (b) 2D advection, (c) 2D advection–diffusion, and (d) 3D advection–diffusion. The κ=0\kappa=0 mode is retained only for advection–Allen–Cahn (Section 3.3 ).
Figure 14: Model-capacity control on Burgers. (a) rℓ2\mathrm{r}\ell_{2} error over the rollout for the proposed model and the baselines at matched (solid) and tenfold (dashed) trainable budgets, with parameter counts in the legend. (b) Spectral error, evaluated over κ≥1\kappa\geq 1 (Section 3.3 ).

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