Geographic Implicit Neural Representations (INRs)

• Geographic INRs are coordinate-based neural fields that represent continuous geospatial signals, offering resolution independence and flexibility across Earth domains.
• They integrate positional encoding and multiscale methods to overcome spectral bias and capture high-frequency features like coastlines and urban boundaries.
• Applications include geophysical inversion, environmental reconstruction, and multimodal geospatial alignment, providing a differentiable framework for complex Earth systems.

Searching arXiv for papers on geographic/geophysical implicit neural representations and related methods. arXiv search query: "geographic implicit neural representation geophysical inversion neural field geospatial" Geographic implicit neural representations (INRs) are coordinate-based neural fields that model geospatial, geophysical, or environmental signals as continuous functions of location rather than as fixed rasters or voxel arrays. In the generic form $$f_\theta : \Omega \subset \mathbb{R}^d \to \mathbb{R}^c$$, the input is a spatial or spatio-temporal coordinate and the output is a scalar field, vector field, occupancy, density, image intensity, or embedding. In Earth settings, $\Omega$ may be planar coordinates, Cartesian $(x,y,z)$, spherical coordinates on $S^2$, or spatio-temporal domains, while $c$ may represent physical variables, semantic labels, or multimodal features. Recent work has instantiated this paradigm for three-dimensional gravity inversion, Earth location encoders, continuous environmental field reconstruction from sparse observations, multimodal geospatial alignment, and distributed volumetric caches, establishing geographic INRs as a broad class of continuous neural representations specialized to geographic structure, sampling, and physics (Essakine et al., 2024, Jayasundara et al., 16 Apr 2026, Mishra et al., 17 Oct 2025, Cai et al., 5 Feb 2025).

1. Formal definition and geographic scope

A geographic INR replaces explicit storage of a field on a grid with a learned function queried at arbitrary coordinates. This is the central formulation in both the INR surveys and Earth-specific work: a signal is represented as $f_\theta : \mathbb{R}^d \to \mathbb{R}^c$, with coordinates mapped directly to signal values or embeddings (Essakine et al., 2024, Jayasundara et al., 16 Apr 2026). In geophysics, a particularly direct example is three-dimensional gravity inversion, where the subsurface density contrast is modeled as a continuous field

$\rho_\theta : \mathbb{R}^3 \to \mathbb{R},$

with $\mathbf{x}=(x,y,z)$ as input and density contrast as output; a voxel model appears only when the field is sampled on a forward-model grid (Mishra et al., 17 Oct 2025). In Earth representation learning, the same idea appears as location encoders of the form

$$f : S^2 \to \mathbb{R}^D,\qquad (\lambda,\phi)\mapsto z,$$

where longitude–latitude coordinates are mapped to high-dimensional embeddings with ambient dimension between 256 and 512 (Rao et al., 3 Nov 2025).

This formulation makes geographic INRs inherently resolution-independent. A trained field can be evaluated at arbitrary coordinates, which is why the environmental reconstruction literature describes them as “resolution-independent” and suitable for sparse, irregular ecological observations, while the signal-processing survey emphasizes that differentiation and related operators can be carried out analytically through automatic differentiation rather than through discrete approximations (Pregowska et al., 20 Apr 2026, Jayasundara et al., 16 Apr 2026). This suggests that geographic INRs are best viewed not merely as compressors of maps, but as function-space models of spatial phenomena.

The coordinate domain is not restricted to regular image grids. Geographic INRs have been applied to lon–lat fields, Cartesian subsurface volumes, image-plane morphology, and multimodal geospatial feature fields. In the ecological reconstruction setting, the same coordinate-based formalism is used for species distribution reconstruction, spatio-temporal phenology with $(\text{lon},\text{lat},\mathrm{DOY})$, and image-plane leaf morphology with $\Omega$0, all with scalar outputs interpreted through a sigmoid as probabilities or relative occurrence surfaces (Pregowska et al., 20 Apr 2026). In multimodal geospatial representation learning, a remote-sensing image is turned into a continuous feature field $\Omega$1, queried at continuous coordinates inside the image footprint to align overhead and street-view modalities (Liu et al., 20 Mar 2025).

2. Spectral bias, positional encoding, and locality

A central technical issue in geographic INRs is spectral bias: plain coordinate MLPs preferentially learn low-frequency structure and tend to oversmooth sharp transitions and short-wavelength variability. The INR survey identifies this as a defining limitation of vanilla coordinate networks, and the signal-processing perspective frames much of modern INR design as reshaping the approximation space to better represent high-frequency or spatially localized content (Essakine et al., 2024, Jayasundara et al., 16 Apr 2026). In geographic applications, this matters because coastlines, urban boundaries, small islands, mountain ridges, sharp fronts, and subsurface interfaces are high-frequency or localized features relative to continental-scale background variation.

The gravity inversion study provides a concrete geophysical demonstration. There, a sinusoidal positional encoding

$\Omega$2

is applied to each coordinate before the MLP, with the full input $\Omega$3. Without positional encoding, the inversion captures only broad, long-wavelength trends of a Gaussian random field; with encoding, sharp edges and fine textures are much better recovered, and density error and RMS gravity residual are reduced by approximately an order of magnitude (Mishra et al., 17 Oct 2025). This is one of the clearest demonstrations that Fourier-style encodings are not merely architectural refinements but determinants of geological recoverability.

Later work generalizes this spectral-locality argument. FLAIR introduces RC-GAUSS, a band-limited and spatially localized activation, together with Wavelet-Energy-Guided Encoding (WEGE), which uses a DWT-derived energy map to inject region-adaptive frequency information into the INR. The paper explicitly argues that standard INRs lack frequency selectivity, spatial localization, and sparse representations, whereas RC-GAUSS and WEGE allow frequency selection and locality to be learned jointly (Ko et al., 19 Aug 2025). In geographic terms, this directly targets heterogeneous signals in which open ocean or deserts are low-frequency while coastlines, cities, ridges, or localized convection require locally concentrated high-frequency capacity.

Wavelet and multiscale encodings are especially salient on the sphere. FAIR-Earth shows that several existing Earth location encodings systematically underperform on islands and coastlines, and proposes spherical wavelet encodings built from spherical Morlet wavelets: $\Omega$4 Because these encodings are localized in both space and scale, they reduce the strong trade-off between large landmasses and small islands observed with spherical harmonics and other global encodings (Cai et al., 5 Feb 2025). A closely related conclusion appears in the reproducible microscopy benchmark: Haar and Fourier encodings achieve the strongest macro-averaged reconstruction fidelity on held-out columns, about 26 dB, and preserve boundaries more accurately than smoother-bias alternatives, reinforcing the general point that explicit spectral and multiscale encodings better preserve boundary-sensitive structure (Pregowska, 26 Mar 2026).

3. Physics-based geographic INRs and inverse problems

One major branch of geographic INRs is physics-based inversion, where the neural field is optimized through a forward model rather than supervised coordinate–value pairs. In the gravity case, the INR produces a model vector

$\Omega$5

at prism centers, and the predicted data are

$\Omega$6

with $\Omega$7 built from the Nagy (1966) rectangular-prism formula. Training uses a pure data-misfit loss on standardized observations,

$\Omega$8

with no explicit Tikhonov term, no smoothness penalty, and no depth weighting (Mishra et al., 17 Oct 2025). The resulting inversion reconstructs detailed structure and geologically plausible boundaries while using network weights rather than one independent parameter per voxel. This is a prototypical geographic INR: a continuous field, trained directly through a differentiable physical operator.

A second geometric branch uses neural fields as implicit surfaces or boundaries. The phase-transition formulation models a signed density $\Omega$9 and trains it with

$(x,y,z)$0

where $(x,y,z)$1 is a double-well potential and $(x,y,z)$2 enforces boundary consistency around observed surface points. The paper proves that, under the stated scaling of $(x,y,z)$3 and $(x,y,z)$4, minimizers converge to proper occupancy functions in $(x,y,z)$5 and the limit minimizer satisfies the reconstruction constraints while having minimal surface perimeter (Lipman, 2021). Its log transform

$(x,y,z)$6

converges to a signed-distance-like field. For geographic geometry—coastlines, terrain surfaces, glacier fronts, subsurface horizons—this gives an explicit regularization principle: among all reconstructions consistent with the data, the learned surface tends toward minimal perimeter or minimal area.

More broadly, the signal-processing survey formulates inverse problems as

$(x,y,z)$7

where $(x,y,z)$8 is a forward operator and $(x,y,z)$9 may include smoothness, derivative constraints, or physics-informed penalties (Jayasundara et al., 16 Apr 2026). This suggests that geographic INRs should be understood not only as interpolants, but as parameterizations of unknown fields inside variational inverse problems.

4. Regionalization, multimodal alignment, and amortized inference

A second development is the move from single global neural fields to structured geographic INRs that model regional heterogeneity explicitly. Superpixel-informed INR replaces pointwise modeling by generalized superpixels, defined as spatially connected clusters obtained by a coordinate-aware k-means++ variant. Each region $S^2$0 has its own INR $S^2$1, while a shared dictionary $S^2$2 ties regions together: $S^2$3 The weather completion experiments, using precipitation, soil water evaporation, vapor pressure, Tmax, and Tmin at three North American locations, show that this region-based representation improves substantially over decision trees, k-nearest neighbors, random forests, and SIREN, with S-INR achieving the best NRMSE and $S^2$4 at all three locations (Li et al., 2024). The technical message is that geographic coherence can be encoded explicitly through region-wise subnetworks and shared latent bases, rather than left entirely to a single global MLP.

A different form of geographic structuring appears in multimodal geospatial foundation models. GAIR introduces an INR module that turns an overhead remote-sensing image into a continuous feature field $S^2$5, queried at the street-view geolocation $S^2$6. The queried remote-sensing embedding is constructed by feature unfolding plus local-ensemble interpolation: $S^2$7 This geographically aligned embedding is then trained with contrastive losses against the colocated street-view embedding and the location encoder (Liu et al., 20 Mar 2025). The paper reports gains across remote-sensing, street-view, and location benchmarks, indicating that continuous queryable overhead fields can function as the geometric glue between modalities. In this setting, a geographic INR is not reconstructing a scalar field in physical units but producing a continuous semantic feature field over geographic coordinates.

Amortized function inference further expands the concept. Versatile Neural Processes (VNP) learn a distribution over INRs conditioned on partial observations $S^2$8, using a bottleneck encoder that converts context sets into a small number of informative tokens and a hierarchical latent-modulated decoder. The target conditional takes the form

$S^2$9

with an ELBO that decomposes KL terms across multiple latent blocks (Guo et al., 2023). Because the set tokenizer operates on unordered context sets, this framework is directly compatible with sparse and irregular geospatial observations. This suggests an important distinction: some geographic INRs are per-instance optimization procedures, while others are meta-learned conditional priors that infer a continuous field in a single forward pass.

5. Scalability, diagnostics, and evaluation on Earth domains

Scalability has become a first-order issue as geographic INRs move from isolated tiles to large distributed fields. Distributed Volumetric Neural Representation (DVNR) shows that local neural fields can be trained independently on distributed subdomains in HPC settings, with no parameter synchronization across ranks. Each MPI rank normalizes coordinates and values locally, trains its own hash-grid-plus-MLP field, and uses ghost cells together with a boundary-focused loss to suppress seams across partitions (Wu et al., 2023). For time-varying fields, weight caching initializes each timestep from the previous one and can reduce training time per timestep by up to $c$0. Although developed for scientific visualization, the method directly suggests tiled or rank-local geographic INR caches for atmosphere, ocean, or land-surface models.

Evaluation methodology has also broadened beyond global error. FAIR-Earth introduces subgroup-aware assessment of geographic INRs on a unified 0.1°×0.1° global grid, including land–sea boundaries, CO$c$1 emissions, precipitation, surface air temperature, and population density. The central finding is that islands and coastlines are systematically worse modeled than large landmasses and open ocean, and that some encodings exhibit negative correlations between mainland and island performance as sampling increases (Cai et al., 5 Feb 2025). This reframes geographic INRs as potentially unfair representations of Earth signals rather than uniformly continuous models.

A related diagnostic is intrinsic dimension. For Earth location encoders $c$2, the intrinsic-dimension study finds that, although the ambient dimension lies between 256 and 512, the intrinsic dimension falls roughly between 2 and 10 and varies with positional encoding resolution and input modalities during pre-training (Rao et al., 3 Nov 2025). Global intrinsic dimension correlates positively with downstream task performance, while local intrinsic-dimension maps reveal pre-training coverage bias and encoding artifacts. Together with subgroup fairness analysis, this introduces a new evaluative vocabulary for geographic INRs: not only interpolation error, but also where complexity is concentrated and who is poorly represented.

Spatially structured validation is equally important. In environmental field reconstruction from sparse ecological observations, blocked cross-validation is used to reduce spatial leakage: 2D tasks are partitioned into $c$3 blocks, and spatio-temporal phenology adds 30-day bins in time (Pregowska et al., 20 Apr 2026). This directly addresses a common misconception that high performance under random splits implies genuine geographic generalization.

6. Limitations, controversies, and research directions

Geographic INRs are not uniformly superior to classical discrete representations. The most explicit challenge comes from the systematic benchmark “Grids Often Outperform Implicit Neural Representations,” which finds that, for most tasks and signals, a simple regularized grid with interpolation trains faster and to higher quality than any INR with the same number of parameters, and that INRs outperform grids mainly when fitting signals with underlying lower-dimensional structure such as shape contours (Kim et al., 10 Jun 2025). For geographic practice, this means that smooth or bandlimited fields on regular grids remain strong territory for explicit discretizations, especially when combined with TV-like regularization.

Even when INRs are effective, they do not remove fundamental ill-posedness. The gravity inversion paper states explicitly that non-uniqueness persists, and that choices of positional encoding bandwidth, network capacity, output bounds, optimizer settings, normalization, and forward discretization constitute an implicit prior that must be calibrated carefully against data resolution and noise (Mishra et al., 17 Oct 2025). FAIR-Earth similarly shows that encoding choice can induce systematic geographic bias, and the signal-processing survey emphasizes unresolved issues of stability, weight-space interpretability, and large-scale generalization (Cai et al., 5 Feb 2025, Jayasundara et al., 16 Apr 2026).

Uncertainty remains underdeveloped in many Earth applications. The gravity inversion framework is deterministic and offers a single point estimate; the ecological field reconstruction work likewise notes the absence of calibrated uncertainty and identifies Bayesian INRs and ensembles as future work (Mishra et al., 17 Oct 2025, Pregowska et al., 20 Apr 2026). This suggests a practical division of labor: current geographic INRs are often strongest as representation layers or differentiable parameterizations inside larger pipelines, rather than as complete probabilistic geospatial models.

The most plausible near-term directions are already visible across the literature: spherical or manifold-aware encodings for global domains, localized multiscale bases for coastlines and heterogeneous regions, physics-informed forward operators for geophysical inversion, distributed tiling for very large volumes, and multimodal continuous fields that align Earth observations across views and sensors (Essakine et al., 2024, Jayasundara et al., 16 Apr 2026, Liu et al., 20 Mar 2025). Taken together, these works indicate that geographic INRs are evolving from a generic coordinate-MLP recipe into a domain-specific family of neural fields whose success depends on matching approximation space, geometry, and inductive bias to the structure of Earth signals.