Physics-Informed Spatio-Temporal Surrogates

• PISTM is a surrogate modeling approach that embeds physical principles to efficiently predict evolving fields governed by nonlinear PDE dynamics.
• Methods range from Koopman autoencoders and explicit PDE residuals to fixed architectural encodings and statistical regressions, each balancing speed and accuracy.
• Applications show that PISTM can reduce simulation runtimes by orders of magnitude while ensuring stability and generalization across diverse operating conditions.

Searching arXiv for the cited papers to ground the article in current literature. Physics-Informed Spatio-Temporal Surrogate Modeling (PISTM) denotes a class of surrogate-modeling approaches for nonlinear spatio-temporal dynamical systems in which the surrogate is constrained or structured by physical principles rather than trained as a purely generic function approximator. In the formulation presented by “Non-intrusive Learning of Physics-Informed Spatio-temporal Surrogate for Accelerating Design” (Mondal et al., 15 Apr 2026), PISTM is a non-intrusive framework for rapidly emulating nonlinear spatio-temporal dynamical systems—demonstrated on two-dimensional incompressible flow around a cylinder—while retaining key physics-induced stability and generalization properties. Across the literature, the term encompasses distinct mechanisms for injecting physics into a surrogate: latent linear dynamical structure via Koopman theory, explicit PDE residuals and hard boundary-condition imposition, discretized PDE operators embedded in the architecture, multi-scale physics-encoded recurrence, and PDE-regularized statistical regression (Mondal et al., 15 Apr 2026, Ren et al., 2022, Chen et al., 8 Apr 2025, Wan et al., 13 Mar 2025, Li et al., 2024).

1. Concept and scope

PISTM addresses settings in which the target quantity is a field or collection of fields evolving over space and time under PDE-governed dynamics. The central motivation is that high-fidelity simulations are often high-fidelity in nature, and can be computationally very expensive, so generating training or evaluation data becomes a bottleneck in design optimization, uncertainty quantification, or parametric studies (Mondal et al., 15 Apr 2026). In this context, the surrogate approximates a map from design variables, operating conditions, coarse simulations, or past observations to future spatio-temporal fields.

The literature represented here uses the term in more than one technical sense. In (Mondal et al., 15 Apr 2026), physics-informed means “informed by the physics of dynamical systems in general” through Koopman theory and stability constraints, with no explicit PDE residual term in the loss. In “Physics-informed Deep Super-resolution for Spatiotemporal Data” (Ren et al., 2022), physics-informed means explicit PDE residual loss computed on reconstructed high-resolution fields using finite-difference operators, plus hard imposition of boundary conditions. In “PINP: Physics-Informed Neural Predictor with latent estimation of fluid flows” (Chen et al., 8 Apr 2025), the PDE itself is embedded into the prediction operator through a hard-coded discrete update and auxiliary residual constraints. In “PIMRL: Physics-Informed Multi-Scale Recurrent Learning for Spatiotemporal Prediction” (Wan et al., 13 Mar 2025), physics is encoded through fixed convolutional operators, periodic padding, and a multi-scale recurrent architecture rather than through PINN-style residual penalties.

This suggests that PISTM is better understood as a family of design patterns rather than a single method. A plausible implication is that what unifies these methods is not a specific loss construction, but the requirement that the surrogate’s latent dynamics, update rule, regularization, or representation be tied to physically meaningful structure.

2. Non-intrusive Koopman-based PISTM

The framework introduced in (Mondal et al., 15 Apr 2026) combines three components: Koopman autoencoders, a convolutional autoencoder reduced-order model, and Gaussian process regression. Its explicit goal is a physics-informed surrogate that does not require explicit PDEs and still generalizes well outside the training conditions.

For each operating condition in the training set, a Koopman autoencoder learns a reduced-order, linear-in-time representation of the dynamics from historical time series and forecasts into a future time window (Mondal et al., 15 Apr 2026). The learned latent dynamics follow the Koopman assumption

$z_{t+1} = K z_t,$

with encoder $\Phi$, decoder $\Psi$, approximate forward–backward consistency, and approximate invertibility. The paper links this construction to consistent Koopman autoencoders, where latent dynamics are constrained to be linear and invertible, and states that these constraints encode generic properties of dynamical systems, such as stability, without explicit PDEs (Mondal et al., 15 Apr 2026).

The second stage does not learn a mapping from operating conditions to the Koopman operator matrix $K$ directly. Instead, Koopman-predicted spatio-temporal fields across all training conditions are stacked and compressed by a convolutional autoencoder into latent codes $z(t)$ (Mondal et al., 15 Apr 2026). The third stage fits Gaussian Process models mapping operating conditions and time to those latent codes,

$\mathcal{R} : (X_{in}, t) \mapsto z,$

so that inference at a new operating condition is performed by GP prediction in latent space followed by decoding (Mondal et al., 15 Apr 2026).

The framework is non-intrusive in the precise sense that it uses only input–output data from an existing solver or experiments; it does not require access to simulation code internals, explicit governing equations, PDE residual computations, or adjoints (Mondal et al., 15 Apr 2026). The physics-informed aspect arises through the structural assumptions of the Koopman operator, the fact that training data are produced by a physically valid high-fidelity solver, and stability constraints demonstrated in prior Koopman autoencoder work.

On two-dimensional incompressible flow around a cylinder, with Reynolds number $Re\in(50,800)$, 45 training samples, 181 historical snapshots per sample, and 10 future prediction steps, the method is evaluated on five unseen Reynolds numbers $Re = 83,172,218,406,594$ (Mondal et al., 15 Apr 2026). The paper defines three relative $L^2$ errors,

$$\varepsilon_E(t),\quad \varepsilon_{KE}(t),\quad \varepsilon_K(t),$$

for emulation-versus-truth, emulation-versus-Koopman, and Koopman-versus-truth, respectively. It reports that $\Phi$0 remains roughly constant over the prediction horizon, and for all test cases except $\Phi$1, $\Phi$2 (Mondal et al., 15 Apr 2026). The high error at $\Phi$3 is attributed to sparse training data in the range $\Phi$4, where only 2 training samples are available.

The same study reports high-fidelity simulation cost of $\Phi$5 minutes per test case, PISTM inference cost of $\Phi$6 seconds, and speedup of $\Phi$7 in testing (Mondal et al., 15 Apr 2026). This suggests that, within its demonstrated regime, the main contribution is not merely reduced runtime, but reduced runtime while preserving the time-stable forecasting behavior associated with the Koopman component.

3. Explicit PDE-constrained formulations

A second major interpretation of PISTM uses explicit PDE residuals and hard boundary-condition enforcement. In (Ren et al., 2022), the task is spatio-temporal super-resolution: given low-resolution fields $\Phi$8, recover high-resolution fields $\Phi$9 so that reconstruction is consistent with coarse data and approximately satisfies the governing PDE. The paper writes the general PDE as

$\Psi$0

and defines a physics loss from a residual

$\Psi$1

with total objective

$\Psi$2

Here $\Psi$3 is an $\Psi$4 data loss on high-resolution labels and $\Psi$5 is a Frobenius-norm PDE residual loss computed using finite-difference kernels (Ren et al., 2022).

That framework exploits the fact that temporal derivative $\Psi$6 and spatial operators act differently in a PDE, and therefore factorizes the surrogate into a temporal module and a spatial module. Temporal interpolation and ConvLSTM refine low-resolution dynamics; a residual CNN with PixelShuffle reconstructs high-resolution spatial detail (Ren et al., 2022). Boundary conditions are hard imposed: Dirichlet conditions by overwriting predicted boundary values, Neumann conditions by ghost nodes used in finite-difference derivative calculations, and periodic conditions by circular padding (Ren et al., 2022).

The method is tested on 2D Rayleigh–Bénard convection, 2D Gray–Scott reaction–diffusion, and 3D Gray–Scott reaction–diffusion. For 2D RBC, reported relative full-field error is $\Psi$7, compared with $\Psi$8 for MeshfreeFlowNet and $\Psi$9 for interpolation; for 2D GS, $K$0 versus $K$1 and $K$2; for 3D GS, $K$3 while MeshfreeFlowNet is out-of-memory (Ren et al., 2022). The same paper reports PhySR parameter counts of $K$4M–$K$5M, 2D RBC inference time of $K$6 ms per time step, and training time per epoch of $K$7 s (Ren et al., 2022).

In (Chen et al., 8 Apr 2025), explicit physics enters even more directly. The model infers a latent physical state $K$8 from past observations of a passive scalar field $K$9, and then advances $z(t)$0 through a hard-coded discrete advection–diffusion update

$z(t)$1

The latent velocity and pressure fields are constrained by incompressible Navier–Stokes residuals

$z(t)$2

and the model also imposes a hinge-style temporal consistency loss on the intermediate scalar state $z(t)$3 (Chen et al., 8 Apr 2025).

The paper evaluates long-horizon prediction on 2D fluid, 2D smoke, 3D smoke, and SEVIR radar data. On synthetic systems it reports lowest MAE/MSE among compared baselines, including Fluid 2D MAE $z(t)$4 versus next best $z(t)$5, and states that PINP recovers latent velocity and pressure with high correlation ($z(t)$6) to ground truth (Chen et al., 8 Apr 2025). On SEVIR nowcasting it reports best CSI-M $z(t)$7 and best high-intensity CSI values at thresholds 219, 181, 160, and 133, despite slightly higher MSE than some baselines (Chen et al., 8 Apr 2025). A plausible implication is that explicit transport structure can improve dynamically important event prediction even when it does not minimize pointwise MSE.

4. Multi-scale and architectural physics encoding

PISTM can also be realized through architectural hard-coding of known operators and multi-scale schedules. In (Wan et al., 13 Mar 2025), the Physics-Informed Multi-Scale Recurrent Learning framework combines a micro-scale module and a macro-scale module. The micro module is a PeRCNN-based $z(t)$8-block approximating the PDE right-hand side and performing a forward Euler step,

$z(t)$9

while known differential terms can be inserted as fixed convolution kernels, referred to as “physics-based FD Conv” (Wan et al., 13 Mar 2025). Periodic padding hard-encodes periodic boundary conditions, and the macro module is a residual ConvLSTM encoder–decoder learning coarse latent dynamics over larger time steps $\mathcal{R} : (X_{in}, t) \mapsto z,$0.

The training procedure separates micro-scale pretraining on finely sampled short trajectories from joint training on coarse-scale long trajectories, with the micro module informing the macro module through message passing (Wan et al., 13 Mar 2025). Unlike PINN formulations, the training loss is plain MSE on rollouts; the physical structure is carried by fixed convolutions, periodic padding, and the integration-like architecture rather than an explicit PDE residual penalty.

The framework is evaluated on 1D KdV, 2D Burgers, 2D FitzHugh–Nagumo, and 2D and 3D Gray–Scott systems. The paper reports average improvements of over $\mathcal{R} : (X_{in}, t) \mapsto z,$1 in both RMSE and MAE evaluation metrics, with maximum enhancements reaching up to $\mathcal{R} : (X_{in}, t) \mapsto z,$2, and notes that in 2D Gray–Scott PIMRL attains HCT $\mathcal{R} : (X_{in}, t) \mapsto z,$3 s versus $\mathcal{R} : (X_{in}, t) \mapsto z,$4 s for PeRCNN and $\mathcal{R} : (X_{in}, t) \mapsto z,$5 s for FNO-coarse, while FNO diverges (NaN) (Wan et al., 13 Mar 2025).

This suggests that in data-scarce regimes, hard architectural encoding of local physics plus a coarse recurrent model can be more robust than either pure neural operators or pure physics-encoded micro-solvers alone.

5. Statistical and semiparametric formulations

Not all PISTM instances are neural surrogates. In “Physics-encoded Spatio-temporal Regression” (Li et al., 2024), the surrogate is restricted to the solution manifold of a linear evolution equation

$\mathcal{R} : (X_{in}, t) \mapsto z,$6

with solution represented through the semigroup

$\mathcal{R} : (X_{in}, t) \mapsto z,$7

Using an eigen-expansion of the spatial operator, the model becomes

$\mathcal{R} : (X_{in}, t) \mapsto z,$8

so estimation reduces to linear regression for the coefficients $\mathcal{R} : (X_{in}, t) \mapsto z,$9 (Li et al., 2024). Physics is encoded as a hard constraint through the solution basis itself, not through a penalty. The paper proves convergence rate

$Re\in(50,800)$0

with optimal choice $Re\in(50,800)$1 yielding

$Re\in(50,800)$2

and establishes a matching minimax lower bound (Li et al., 2024). This provides a rare information-theoretic characterization of a physics-encoded spatio-temporal surrogate.

A related but distinct approach appears in “Modeling group heterogeneity in spatio-temporal data via physics-informed semiparametric regression” (Sanctis et al., 17 Nov 2025). There, the nonparametric component $Re\in(50,800)$3 is regularized by a PDE-based penalty

$Re\in(50,800)$4

where $Re\in(50,800)$5 is a linear second-order elliptic advection–diffusion–reaction operator, and a temporal penalty

$Re\in(50,800)$6

is added (Sanctis et al., 17 Nov 2025). The resulting semiparametric mixed-effects model separates a common physics-informed field from group-specific random effects. In simulation, MEST-PDE achieves the lowest RMSE for the nonparametric field and better fixed-effect estimation than thin-plate-spline and soap-film alternatives (Sanctis et al., 17 Nov 2025). On hourly NO$Re\in(50,800)$7 data in Lombardy, the model uses wind-driven advection–diffusion regularization and random effects for sensor-technology heterogeneity (Sanctis et al., 17 Nov 2025).

These formulations show that PISTM can be cast as a reduced-basis statistical inverse problem or a PDE-regularized mixed model, not only as a neural operator or PINN.

6. Applications, performance, and limitations

The applications represented in this literature are broad but share the same computational logic: replace repeated or long-horizon high-fidelity PDE solves with a faster surrogate that remains physically plausible.

The Koopman–ROM–GP framework in (Mondal et al., 15 Apr 2026) targets accelerating design for 2D cylinder wake flow and emphasizes rapid evaluation for unseen operating conditions. The super-resolution framework in (Ren et al., 2022) targets coarse-to-fine reconstruction across fluid and reaction–diffusion systems without sacrificing PDE consistency. PINP (Chen et al., 8 Apr 2025) targets fluid and scalar forecasting, including real-world extreme-precipitation nowcasting. PIMRL (Wan et al., 13 Mar 2025) targets long-term PDE prediction across one to three dimensions. Statistical and semiparametric variants address wildfire front propagation (Dabrowski et al., 2022), advection–diffusion processes with Gibbs-phenomenon suppression (Wei et al., 2022), and environmental monitoring with group heterogeneity (Sanctis et al., 17 Nov 2025). A graph-based hybrid GNN–LSTM variant uses physical node features from a regional climate model to predict deep polar ice layer thickness (Liu et al., 2024). A transformer-health application uses a 1D heat-diffusion PINN with Residual-Based Attention to reconstruct oil temperature, winding temperature, and ageing fields (Ramirez et al., 2024).

Several recurrent limitations also appear across these works. Coverage of parameter space matters: in (Mondal et al., 15 Apr 2026), performance at $Re\in(50,800)$8 degrades because the range $Re\in(50,800)$9 contains only 2 training samples. Explicit PDE-residual methods depend on the fidelity of the governing PDE and boundary specification; (Mondal et al., 15 Apr 2026) avoids this dependency by foregoing explicit PDEs, but consequently does not guarantee PDE residual satisfaction. Hard-coded explicit schemes such as in (Chen et al., 8 Apr 2025) may become unstable for high Pe/Re if extended without more sophisticated time integration. Multi-scale recurrent methods such as (Wan et al., 13 Mar 2025) require known PDE terms to exploit fixed convolutional operators fully. Spectral and semiparametric approaches typically rely on linear or linearized operators and may not readily cover strongly nonlinear multi-physics without reformulation (Li et al., 2024, Sanctis et al., 17 Nov 2025).

A common misconception is that “physics-informed” necessarily means adding a PINN residual term to a neural network loss. The surveyed work contradicts that. In (Mondal et al., 15 Apr 2026), there is no explicit PDE residual term. In (Wan et al., 13 Mar 2025), physics is active through hard-coded differential operators and periodic padding. In (Li et al., 2024), the surrogate is confined to the exact PDE solution family. In (Ren et al., 2022) and (Chen et al., 8 Apr 2025), by contrast, residual penalties and hard discrete updates are central. Physics-informed spatio-temporal surrogate modeling is therefore best understood as a spectrum of methods that use physics either as a hard constraint, a soft regularizer, a latent dynamical prior, an architectural bias, or a structured output relation.

Taken together, these studies indicate that the technical core of PISTM lies in how physical structure is embedded into the surrogate’s hypothesis class. This suggests that future work will likely continue to hybridize these strategies: non-intrusive latent dynamics for black-box settings, explicit residual or discrete-operator enforcement when equations are available, multi-scale schedules for long-horizon stability, and probabilistic or statistical formulations when uncertainty quantification and hierarchical heterogeneity are central.