Spatial-Physics Informed Models

• SPIM is a modeling approach that represents target fields as continuous functions of spatial coordinates while embedding physics constraints like PDEs and boundary conditions.
• It employs diverse mechanisms such as soft residual enforcement, hard architectural encoding, and latent physics priors to improve tasks ranging from acoustic upsampling to image deblurring.
• The framework has demonstrated enhanced performance metrics (e.g., NMSE, PSNR, SSIM) across applications while addressing challenges in identifiability and boundary treatment.

Searching arXiv for the cited SPIM-related papers to ground the article. Spatial-Physics Informed Model (SPIM) denotes an emerging class of models in which the target of inference is treated as an explicitly spatial field and physical structure is embedded into learning, reconstruction, or prediction. Across recent work, the term is used both as a broad recipe for coordinate-continuous or region-aware field modeling and, in at least one application, as the explicit name of a three-stage workflow for SQUID microscopy data processing. Common to these uses is the replacement of purely black-box interpolation by spatial representations constrained by governing equations, boundary behavior, conservation laws, or reduced physical surrogates (Miotello et al., 2024, Senthilnath et al., 16 Jul 2025, Likhite et al., 9 Nov 2025).

1. Concept and scope

A SPIM is not a single architecture. In the broad sense articulated for spherical microphone array upsampling, the transferable pattern is to represent the target field as a continuous function of spatial coordinates via an implicit neural network, use sparse physical measurements only where sensors exist, impose the governing PDE in the ambient domain through differentiable residual penalties, adopt activations capable of representing multiscale or high-frequency structure, and query the trained model at arbitrary spatial coordinates to obtain dense reconstructions or virtual sensor outputs (Miotello et al., 2024). In image restoration, an analogous pattern appears as a PDE-based global feature layer inserted at the bottleneck of encoder-decoder networks, where latent features are evolved by an advection-diffusion equation rather than by unconstrained local convolutions alone (Likhite et al., 9 Nov 2025). In stochastic field modeling, the same idea takes the form of a learned spatial basis constrained by the governing differential equation and boundary conditions, combined with a generative model over basis coefficients (Zhou et al., 23 Mar 2025).

This range suggests that SPIM is better understood as a modeling principle than as a fixed network family. The spatial object may be a coordinate-based neural field, a latent feature tensor, a graph over irregular entities, a partition-of-unity decomposition, or a basis expansion. The physics may enter as a hard-encoded operator, a soft residual loss, a boundary-aware parameterization, a coarse simulator used as conditioning, or a reduced-order constitutive relation. The common denominator is that space is modeled explicitly and physics is used to restrict admissible reconstructions or dynamics rather than being appended only as post hoc interpretation.

The literature also shows that SPIM can target markedly different tasks: spatial upsampling of spherical microphone arrays, motion deblurring, geostatistical inversion, spatiotemporal super-resolution, stochastic PDE solution generation, flood-depth mapping, rainfall forecasting, current reconstruction from SQUID microscopy, and static-snapshot inference of spatial stochastic dynamics (Miotello et al., 2024, Zheng et al., 2019, Ren et al., 2022, Zhou et al., 23 Mar 2025, Wu et al., 12 Nov 2025, Devda et al., 2024, Senthilnath et al., 16 Jul 2025, Gu et al., 2 Jul 2026).

2. Mathematical formulations

One canonical SPIM formulation is the coordinate-based neural field. In spherical microphone array upsampling, the unknown acoustic pressure on the sphere is represented as

$$\tilde p(\mathbf r_s,k)=f_{\boldsymbol\theta}(\mathbf r_s)\approx p(\mathbf r_s,k),$$

and training combines data fidelity at microphone positions with a Helmholtz residual at collocation points: $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$ Because the pressure is complex-valued, the implementation separates real and imaginary parts into two parallel subnetworks, and the loss includes both reconstruction error and PDE violation (Miotello et al., 2024).

A second formulation places the physics in latent space rather than output space. In motion deblurring, the governing PDE for feature evolution is

$$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$

where diffusion smooths, advection transports information directionally, and the source term injects evidence from degraded features. The PDE is discretized into differentiable iterative updates and inserted at the bottleneck of restoration networks, yielding a low-cost global propagation mechanism (Likhite et al., 9 Nov 2025).

A third formulation is basis-based spatial compression. In the scalable physics-informed deep generative model for stochastic differential equations, the field is represented as

$$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$

where $\phi_i(\bm{x})$ are learned spatial basis functions and $\psi_i(\mathcal{U})$ are coefficients conditioned on stochastic inputs or observations. A generative model then learns the coefficient distribution, and new field samples are reconstructed by the inner product of generated coefficients with the physics-informed basis (Zhou et al., 23 Mar 2025).

A fourth formulation hard-encodes the PDE solution class itself. In physics-encoded spatio-temporal regression, the field is assumed to satisfy

$u' + \mathscr L u = 0,$

with solution

$$u(x,t)=e^{-t\mathscr L}g_0(x)=\sum_{k=1}^\infty \alpha_k e^{-\lambda_k t}\psi_k(x).$$

The statistical task is then reduced to estimating the initial spatial state $g_0$, or equivalently the mode coefficients $\alpha_k$, inside a family of functions that already satisfies the PDE exactly (Li et al., 2024).

These formulations differ in representation, but they share a structural decomposition into spatial object, physical operator, and inference mechanism. This suggests that SPIM is naturally expressed as a constrained field model rather than as a generic predictor on flattened observations.

3. Mechanisms for embedding physics into space

Recent work exhibits several distinct mechanisms for injecting physics into a spatial model.

Soft residual enforcement appears in classical PINN-like SPIMs. In spherical acoustics, the Helmholtz equation is enforced at collocation points through automatic differentiation, with a tiny empirical weight $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$0 on the PDE term because second-order derivatives make the residual numerically sensitive (Miotello et al., 2024). In spatiotemporal super-resolution, PhySR combines an $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$1 data loss with a PDE residual loss computed from finite-difference temporal and spatial derivative filters, while boundary conditions are hard-imposed by explicit padding, ghost nodes, or direct insertion of boundary values (Ren et al., 2022).

Hard architectural encoding replaces residual penalties by built-in operator structure. PIMRL’s micro-scale module is based on PeRCNN and can embed known PDE terms directly as finite-difference convolution kernels, while periodic boundary condition padding is used in both micro and macro modules (Wan et al., 13 Mar 2025). The Boundary-Informed Method of Lines for PINNs similarly replaces fixed finite-difference stencils by a neural network representing the spatial profile, then uses automatic differentiation to compute spatial derivatives for a MOL-generated ODE system, with a secondary temporal PINN replacing classical time integration (Cederholm et al., 17 Oct 2025).

Physics-shaped latent priors are central in generative SPIMs. In physics-informed semantic inpainting for geostatistics, a WGAN-GP generator produces four-channel fields $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$2, while Darcy’s law, mass balance, and boundary conditions are imposed through Sobel-filter-based residual penalties during generator training. Conditioning on sparse direct and indirect measurements is then performed by latent optimization rather than by retraining the generator (Zheng et al., 2019).

Physics as conditioning rather than constraint appears in PIFF. There, a simplified inundation model is computed from DEM and rainfall, and the resulting raster prior is concatenated with the intermediate model state $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$3 during both training and sampling. The physics is not introduced through a separate residual penalty; it is injected as a spatially aligned field-valued prior (Wu et al., 12 Nov 2025).

Spatial decomposition of physics parameters is the defining move in POU-PINNs. A partition-of-unity network outputs weights $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$4 with $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$5, and the PDE coefficient field is represented as

$\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$6

This yields an unsupervised domain decomposition in which subdomains and region-wise parameters are discovered jointly through the physics residual (Rodriguez et al., 2024).

Spatially organized training can itself function as a SPIM component. In curriculum learning based on spatial correlation, the domain is partitioned into subregions and then layered by distance to the boundary. The layer weights

$\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$7

create a boundary-to-interior curriculum, a low-frequency bridge fits pseudo-labels at anchor points to suppress global drift, and region-adaptive reweighting uses local residuals together with gradient norms to target under-optimized subregions (Chen et al., 14 May 2026).

4. Representative applications

The concept has been instantiated across acoustics, vision, geoscience, stochastic PDEs, hydrology, and sensing. The applications are methodologically diverse, but each uses an explicit spatial representation together with physically structured supervision or regularization.

Application	Spatial representation	Reported result
Spherical microphone array upsampling	Implicit neural representation of acoustic pressure on a sphere	Proposed model outperforms SARITA and SIREN variants; at $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$8, mean NMSE is $\nabla^2 p(\mathbf r,k)+k^2 p(\mathbf r,k)=0.$9 dB versus $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$0 dB for SIREN+PDE and $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$1 dB for SARITA
Motion deblurring	PDE-based global feature layer at bottleneck	Adds only about $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$2 extra inference GMACs and improves PSNR and SSIM across FFTformer, NAFNet, Restormer, and Stripformer
Geostatistical inversion	WGAN-GP prior over multi-channel spatial fields	Mean RMSE $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$3 and mean SSIM $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$4 between generated and MODFLOW head fields across 10,000 samples
Spatiotemporal super-resolution	ConvLSTM temporal module plus spatial SR network with PDE residual	On 2D RBC, error decreases from $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$5 for MeshfreeFlowNet to $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$6 for PhySR
Stochastic SDE solution generation	Physics-informed spatial basis plus GAN over coefficients	Demonstrated on a forward problem with 38-dimensional stochastic space and 20-dimensional spatial space
Flood depth mapping	Conditional flow model conditioned on DEM, rainfall, and SPM raster	Average generation time per image is $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$7 s for PIFF versus $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$8 s for TUFLOW

In acoustics, the SPIM pattern is particularly explicit. Starting from low-order spherical microphone arrays with $$\frac{\partial H}{\partial t} = \nabla \cdot ( D \nabla H ) - \nabla \cdot ( \mathbf{v} H ) + f(I),$$9, the model learns a continuous map from spherical coordinates to complex pressure and enforces the Helmholtz equation at collocation points. On measured room impulse responses from the Eigenmike EM32, the proposed Rowdy-SIREN + PDE model gives the best mean NMSE across all tested $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$0 values, with gains over plain SIREN ranging from $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$1 dB to $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$2 dB and gains over SARITA ranging from $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$3 dB up to $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$4 dB (Miotello et al., 2024).

In image restoration, the emphasis shifts from output-space physics to latent-space propagation. The advection-diffusion layer is inserted at the bottleneck of encoder-decoder models and trained progressively with $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$5, $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$6, and $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$7 iterations while keeping total integration time $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$8. Direct training with fixed $$u_{\mathcal{NN}(\mathcal{U},\bm{x})} = \sum_{i=1}^{p}\psi_i(\mathcal{U})\phi_i(\bm{x}),$$9 diverged at epoch 18, whereas the progressive schedule converged and improved NAFNet to PSNR $\phi_i(\bm{x})$0 on GoPro, $\phi_i(\bm{x})$1 on RealBlur-R, and $\phi_i(\bm{x})$2 on RealBlur-J (Likhite et al., 9 Nov 2025).

In geostatistics, SPIM takes the form of a learned prior over realistic geological and hydraulic fields. The unknown hydraulic conductivity field is inferred from sparse direct $\phi_i(\bm{x})$3 measurements and indirect hydraulic-head measurements, with physics embedded through Darcy-flux and divergence residuals. The paper’s benchmark is described as a “high-dimensional problem with 512 dimensions,” referring to the retained KL expansion dimension rather than the number of pixels (Zheng et al., 2019).

In spatiotemporal super-resolution, the separation of temporal and spatial structure in the governing PDE leads to a temporally interpolated ConvLSTM module and a residual spatial reconstruction network. The model is physics-informed through finite-difference PDE residuals and hard boundary encoding rather than through a simulator in the loop. On 3D Gray–Scott reaction–diffusion, PhySR achieves $\phi_i(\bm{x})$4 relative error, while MeshfreeFlowNet fails due to memory (Ren et al., 2022).

In flood mapping, the model is generative in formulation but practically deterministic in inference, because a continuous flow is integrated backward from the DEM-side state using an ODE solver. The physics prior is a simplified inundation model raster, and the rainfall sequence is encoded by a transformer and injected by cross-attention. On a $\phi_i(\bm{x})$5 study area in Tainan represented as $\phi_i(\bm{x})$6 rasters at 20 m resolution, PIFF reports L1 $\phi_i(\bm{x})$7, FID $\phi_i(\bm{x})$8, MAE $\phi_i(\bm{x})$9, and MD $\psi_i(\mathcal{U})$0 on real-event rainfall, with $\psi_i(\mathcal{U})$1 s/image inference (Wu et al., 12 Nov 2025).

5. Identifiability, boundary conditions, and failure modes

A major development in the SPIM literature is the shift from asking whether a model can fit a spatial pattern to asking whether the inferred physics is structurally identifiable. For static spatial stochastic dynamics observed through a single Poisson point-pattern snapshot, the steady-state intensity $\psi_i(\mathcal{U})$2 satisfies

$\psi_i(\mathcal{U})$3

The identifiability analysis shows that if the birth rate $\psi_i(\mathcal{U})$4 is a smooth diffuse function rather than a singular point source, then $\psi_i(\mathcal{U})$5 is non-identifiable for any stochastic convention $\psi_i(\mathcal{U})$6. By contrast, a point source $\psi_i(\mathcal{U})$7 can restore identifiability under additional assumptions, including known source location, known boundary conditions, and $\psi_i(\mathcal{U})$8 at the source (Gu et al., 2 Jul 2026).

Boundary conditions are therefore not merely numerical details. The same paper shows that unknown Robin permeability can destroy identifiability even when the source is a point source and $\psi_i(\mathcal{U})$9 is $u' + \mathscr L u = 0,$0 at the source. Inference is also shaped by the stochastic calculus convention because the diffusion operator depends on $u' + \mathscr L u = 0,$1: $u' + \mathscr L u = 0,$2 The Itô, Stratonovich, and Fickian cases differ in the presence or absence of $u' + \mathscr L u = 0,$3 terms, so the convention must come from microscopic or macroscopic physics rather than from the static snapshot alone (Gu et al., 2 Jul 2026).

Training pathologies are equally prominent. In latent PDE modules for motion deblurring, repeated iterative updates cause gradient instability; progressive solver unrolling is introduced precisely because fixed multi-step training diverges (Likhite et al., 9 Nov 2025). In curriculum PINNs, the failure mode is ineffective spatial information propagation, which leads to spurious convergence or trivial solutions unless boundary-adjacent regions are emphasized first and global low-frequency drift is explicitly controlled (Chen et al., 14 May 2026). In POU-PINNs, the paper reports that “The model did not maintain stability due to partition discontinuity, but it led to the correct solution,” indicating that soft spatial decomposition does not automatically eliminate interface-related optimization issues (Rodriguez et al., 2024).

Boundary treatment recurs as a decisive factor across domains. PhySR hard-imposes Dirichlet, Neumann, and periodic conditions because derivative estimates near boundaries become unreliable under naive padding (Ren et al., 2022). In SQUID microscopy, geometric skew and rotation must be corrected before Biot–Savart inversion, otherwise even a correct inversion law acts on a spatially misregistered field image (Senthilnath et al., 16 Jul 2025). In SPIMs for static biological snapshots, the boundary operator itself changes identifiability. This suggests that a practically reliable SPIM must specify not only the governing PDE but also the boundary model, regularity assumptions, and observation process.

6. Terminology, limitations, and research directions

The acronym SPIM is not unique. In optics and analog optimization, SPIM also denotes the spatial photonic Ising machine introduced by Pierangeli et al. and extended through a multicomponent low-rank computing model for arbitrary real symmetric Ising matrices. That SPIM is an optical Ising architecture rather than a spatial-physics informed field model (Yamashita et al., 2023). The collision is terminological rather than conceptual, but it is now part of the literature and can produce ambiguity in indexing and citation.

Within the physics-informed modeling sense, the literature points to several recurring limitations. Many models remain setup-specific or domain-specific. The spherical microphone array method operates “within its domain of application,” uses only the Helmholtz equation, and does not explicitly incorporate spherical boundary conditions or SH-domain modal consistency in the loss (Miotello et al., 2024). The PDE-enhanced deblurring framework is empirically effective but does not provide formal stability conditions, convergence analysis, or explicit boundary-condition specification (Likhite et al., 9 Nov 2025). PhySR still requires paired low-resolution and high-resolution supervision and is limited to structured grids (Ren et al., 2022). PIFF is trained on TUFLOW-generated flood maps rather than direct observations and does not report calibrated predictive uncertainty (Wu et al., 12 Nov 2025). POU-PINNs require the number of partitions to be chosen a priori and do not supply a formal identifiability analysis for the learned decomposition (Rodriguez et al., 2024).

The current direction of travel is nonetheless clear. Multiple papers call for richer physical constraints, more explicit boundary handling, and more demanding evaluation regimes. In acoustics, future work is to incorporate additional physical constraints more closely tied to the application and to test more challenging scenarios (Miotello et al., 2024). In latent PDE restoration, promising directions include anisotropic or nonlinear diffusion, multi-scale PDE integration, and explicit boundary handling (Likhite et al., 9 Nov 2025). In scalable stochastic generative modeling, future work includes replacing GANs with diffusion, flow matching, or score-based models and extending from steady to time-dependent problems (Zhou et al., 23 Mar 2025). In static-snapshot biological inference, the central recommendation is that any SPIM should be accompanied and guided by careful identifiability analysis before the inferred spatial dynamics are given mechanistic interpretation (Gu et al., 2 Jul 2026).

Taken together, these works suggest that SPIM is evolving toward a family of models that are more explicit about spatial representation, more careful about what physical knowledge is hard-coded versus softly enforced, and more attentive to identifiability and boundary structure. A plausible implication is that the most robust future SPIMs will combine several of the currently separate strands: field-level spatial parameterization, physically meaningful latent evolution, domain decomposition or graph structure where geometry is irregular, explicit observation models, and identifiability-aware training objectives.