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

Junfeng Chen

nlin.CDAug 11, 2026

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

Authors: Junfeng Chen

Organizations: Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, China

Abstract

We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, we apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, we propose a spatiotemporal neural operator mapping a resolved history slab on $Ω\times[-T_{\mathrm{in}},0]$ to a resolved future slab on $Ω\times[0,T_{\mathrm{out}}]$ . Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. This causal temporal operator encodes finite-memory effects in the resolved dynamics while preserving the directionality of the history-to-future map. To improve rollout robustness and suppress nonconservative artifacts, we embed a flux-form inductive bias by parameterizing the windowed update in explicit divergence form. We also provide a data-driven guideline for selecting the memory length $T_{\mathrm{in}}$ via the decorrelation time of a closure-injection diagnostic computed from filtered trajectories. We validate on the coarse-grained viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows, obtaining stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.

Figures & tables

Fig. 1 : Effect of spectral coarse-graining on the viscous Burgers’ dynamics.

Fig. 2 : Viscous Burgers’: Relative

L^{2}

error

\varepsilon(t)

in state space during the rollout.

Fig. 3 : Viscous Burgers’: Mode-wise absolute error of resolved energies.

Fig. 4 : Autoregressive rollout on a held-out filtered Burgers’ trajectory.

Fig. 5 : Viscous Burgers’: Evolution of resolved total energy

E(t)

, normalized by

E(0)

Fig. 6 : Viscous Burgers’: Evolution of mode-wise energy

\frac{1}{2}\,\Big|\mathcal{F}[\bar{u}(\cdot,t)]_{k}\Big|^{2}

for

1\leq k\leq k_{\mathrm{cut}}

Fig. 7 : Effect of spectral coarse-graining on the KS dynamics.

Fig. 8 : Autoregressive rollout on the held-out filtered KS segment: reference (left) and prediction (right).

Fig. 9 : Kuramoto–Sivashinsky: (left) relative

L^{2}

rollout error

\varepsilon(t)

; (right) time-averaged resolved energy spectrum

S(k)

Fig. 10 : Two-point statistics of the filtered KS dynamics: time-averaged spatial autocorrelation (left) and temporal autocorrelation of resolved energy fluctuation (right).

Panel of Table 1 — Table 1: Quantitative diagnostics on filtered KS dynamics. The integral length scale $\ell_{\mathrm{int}}$ is defined in ( 46 ), the energy integral time scale $\tau_{E}$ in ( 49 ), and the energy spectrum error $\varepsilon_{S}$ in ( 43 ).

Fig. 11 : Autoregressive rollout on the held-out filtered NS trajectory.

Fig. 12 : Long-time statistical comparison on the held-out filtered NS trajectory. The learned locally conservative model provides the closest agreement with reference across spectra, vorticity PDF, and spatial autocorrelation, while the LES Smagorinsky baseline remains more dissipative and less accurate at the resolved scales.

Fig. 13 : Two-dimensional spatial autocorrelation contours of vorticity. The locally conservative model reproduces the reference contour geometry more accurately than both the globally conservative model and the LES Smagorinsky baseline.

$T_{\mathrm{in}}$	$T_{\mathrm{in}}/\tau_{\mathrm{decc}}$	$\varepsilon(\downarrow)$
$0.125$	$0.63$	$0.3823$
$0.25$	$1.25$	$0.07483$
$0.5$	$2.50$	$\mathbf{0.06196}$
$1.0$	$5.00$	$0.06682$

Table 2: Sensitivity of performance to the memory length

T_{\mathrm{in}}

(all other settings fixed). Left: viscous Burgers (time-averaged relative

L^{2}

error in state space). Right: KS (relative

\ell_{2}

error of the time-averaged resolved energy spectrum treated as a 12-dimensional vector over

k=1,\dots,12

Panel of Table 3 — Table 3: Effect of temporal history representation on filtered KS dynamics: The integral length scale $\ell_{\mathrm{int}}$ , the energy integral time scale $\tau_{E}$ , the energy spectrum error $\varepsilon_{S}$ , the number of trainable parameters, and the models’ training cost.

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Aug 25, 2026J/K move · Enter open · S save

Physics-Integrated Operator Learning via Gaussian Splatting Representations

Jihao Zhang, Junyi Guo, Jian-Xun Wang

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

PDE Operator Learning PDE Surrogate Modeling Physics-Informed Neural Operators

Open paper AskPDF arXiv

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.

Abstract

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