Encode–Approximate–Reconstruct Operator Learning

• The paper introduces the EAR paradigm by factorizing surrogate models into an encoder, approximator, and reconstructor for learning operators between infinite-dimensional function spaces.
• It applies to a variety of architectures such as Fourier Neural Operators, DeepONets, and graph-based models, demonstrating advantages like mesh-independence and resolution invariance.
• The framework provides rigorous error decompositions and universal approximation guarantees, highlighting trade-offs between expressiveness and computational efficiency.

Encode–Approximate–Reconstruct (EAR) operator learning is a unifying perspective for learning maps between infinite-dimensional function spaces. In its standard form, one seeks an operator $G:X\to Y$, typically between separable Banach or Hilbert spaces of functions, from data $\{(u_i,G(u_i))\}_{i=1}^N$. EAR factorizes the learned surrogate into an encoder $e$, an approximator $A$, and a reconstructor $r$, so that

$$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$

with the encoder lifting the input function into latent features, the approximator acting on those features—often through nonlocal layers—and the reconstructor mapping latent features back to the output space (Kovachki et al., 2024). This viewpoint covers kernel neural operators, Fourier Neural Operators, DeepONets, graph-based operator learners, finite-element-based architectures, and several non-neural constructions, and it is equally natural in statistical, approximation-theoretic, and PDE-constrained formulations (Kovachki et al., 2024, Subedi et al., 4 Apr 2025).

1. Formal operator-learning setting

The canonical setting takes $X$ and $Y$ to be separable spaces of functions, for example $X=L^p(\Omega;\mathbb R^m)$ and $Y=L^q(\Omega';\mathbb R^n)$, and studies a target operator $\{(u_i,G(u_i))\}_{i=1}^N$0 with data sampled either from a distribution $\{(u_i,G(u_i))\}_{i=1}^N$1 on inputs or a joint law on input–output pairs (Kovachki et al., 2024). In PDE applications, inputs may encode coefficient fields, forcings, geometries, initial conditions, or boundary data, while outputs are solutions or observables. In the statistical formulation, this becomes function-to-function regression: a hypothesis class $\{(u_i,G(u_i))\}_{i=1}^N$2 is fitted by empirical risk minimization against a loss on the output function space, often $\{(u_i,G(u_i))\}_{i=1}^N$3, relative $\{(u_i,G(u_i))\}_{i=1}^N$4, or Sobolev-type losses (Subedi et al., 4 Apr 2025).

Within EAR, the encoder $\{(u_i,G(u_i))\}_{i=1}^N$5 is a lifting map into a latent space $\{(u_i,G(u_i))\}_{i=1}^N$6, the approximator $\{(u_i,G(u_i))\}_{i=1}^N$7 performs the principal operator approximation, and the reconstructor $\{(u_i,G(u_i))\}_{i=1}^N$8 decodes back into the output space (Kovachki et al., 2024). The latent space may itself be a function space, a coefficient space, a graph feature space, or a finite-dimensional Euclidean space. A central purpose of this factorization is to isolate three distinct error sources: input compression or sensing, approximation in latent coordinates, and output reconstruction. In one formulation, this is expressed by the EAR-aligned decomposition

$\{(u_i,G(u_i))\}_{i=1}^N$9

where $e$0 is an encoding mismatch, $e$1 is finite-dimensional approximation error, and $e$2 is reconstruction error (Kovachki et al., 2024). This decomposition is foundational because it makes explicit that network depth and nonlocality primarily target $e$3, whereas basis quality, projection quality, and decoder richness control the other two terms.

A recurrent misconception is that EAR is specific to one neural architecture family. The literature instead treats it as a structural template: branch–trunk networks, spectral operators, finite-element discretizations, operator-valued RKHS regression, and weighted least-squares constructions all instantiate the same encode–approximate–reconstruct pattern, albeit with different choices of latent representation and approximation mechanism (Kovachki et al., 2024, Ziarelli et al., 2024, Turnage et al., 11 Dec 2025).

2. Architectural realizations

The most widely used EAR realizations are neural operators with nonlocal layers. A canonical integral-kernel layer acts on a latent feature $e$4 by

$e$5

with pointwise channel mixing $e$6, a learnable integral kernel $e$7, bias $e$8, and nonlinearity $e$9 (Kovachki et al., 2024). In EAR terms, the lifting layer plays the role of encoder, stacked kernel layers form the approximator, and a final projection supplies reconstruction.

Fourier Neural Operators replace explicit kernels by spectral convolution: $A$0 where $A$1 is the Fourier transform and $A$2 is a learnable spectral multiplier with mode truncation (Kovachki et al., 2024). The encoder is a lifting map to latent channels, the approximator is the learned multiplier in Fourier space, and reconstruction occurs through inverse transforms inside the layers and a final channel projection. Parameter counts can be independent of discretization via fixed Fourier-mode truncation, which underlies mesh-independence and resolution lifting (Kovachki et al., 2024).

DeepONet realizes EAR through a branch–trunk decomposition,

$A$3

where the branch network serves as a functional encoder, the latent approximation is the bilinear pairing, and the trunk network reconstructs the output function through a learned basis expansion (Kovachki et al., 2024). This architecture is especially natural when sensor values, PCA coefficients, or other low-dimensional summaries are available.

The same decomposition appears in graph and structure-aware settings. In SPIN for spatiotemporal graph imputation, tokens $A$4 are encoded using observed values and spatiotemporal positional embeddings, the approximator is stacked masked spatiotemporal attention on the graph–time product space, and a small MLP readout reconstructs missing values (Marisca et al., 2022). In SPON, finite-element projection/interpolation provides the encoder, a graph neural processor or multigrid processor acts as the approximator, and finite-element synthesis with boundary-condition enforcement acts as the decoder (Bouziani et al., 2024). In kernel operator learning for epidemic trajectories, RKHS-based optimal recovery provides encoding and reconstruction, while vector-valued kernel regression supplies the latent approximation (Ziarelli et al., 2024).

Architecture	EAR realization	Salient note
Integral-kernel neural operator	lifting $A$5 nonlocal kernel layers $A$6 projection	mesh-free nonlocality
Fourier Neural Operator	lifting $A$7 spectral multipliers $A$8 inverse transform and projection	mode truncation yields mesh-independence
DeepONet	branch encoder $A$9 bilinear pairing $r$0 trunk basis reconstruction	flexible sensing and coordinate evaluation
SPIN	masked token embedding $r$1 graph–time attention $r$2 MLP readout	conditions on observed values during propagation
SPON	FE projection $r$3 GNN or multigrid processor $r$4 FE reconstruction	exact Dirichlet enforcement on complex geometries

These realizations also clarify architecture-specific limitations. FNO is strongest on periodic or near-periodic regular grids but requires modifications for complex geometries; DeepONet’s linear reconstruction in a fixed trunk basis may underperform on discontinuities unless enriched; kernel and graph operators demand careful boundary handling and scaling; SPON trades architectural simplicity for explicit discretization structure (Kovachki et al., 2024, Bouziani et al., 2024).

3. Training, discretization, and numerical realization

The standard training objective is empirical risk minimization,

$r$5

with regularization on weights, kernels, or truncation parameters (Kovachki et al., 2024). Optimization is typically performed with stochastic gradient methods, although convex training is available in random-feature operator models (Kovachki et al., 2024).

A central numerical issue is the relation between continuum operators and discrete training data. In the discretization-independent EAR formulation, the encoder is a linear projection onto learned basis functions,

$r$6

the approximator maps $r$7 to output coefficients $r$8, and the reconstructor synthesizes

$r$9

Because reconstruction produces a function in the output space rather than values on a fixed grid, the same learned operator can be evaluated on arbitrary output discretizations, provided the encoder quadrature is accurate on the chosen input discretization (Hauck et al., 9 Jul 2025). This is the paper’s notion of discretization independence, and it is particularly effective for multifidelity learning across mixed resolutions (Hauck et al., 9 Jul 2025).

An alternative non-neural approximator is weighted least squares in a finite-dimensional subspace $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$0. The sampling measure is chosen through an operator-level Christoffel function: $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$1, with weights $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$2. This yields uniformly well-conditioned Gram matrices and near-optimal sample complexity, with the number of training samples $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$3 on the order of $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$4 (Turnage et al., 11 Dec 2025). In EAR terms, the basis evaluations provide the encoding, weighted least squares performs approximation in the latent coefficient space, and reconstruction is the synthesis $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$5 (Turnage et al., 11 Dec 2025).

Practical mesh-independence therefore should not be conflated with architectural rhetoric. In the literature it depends either on continuum-conceived layers whose parameters do not scale with mesh size, or on encoders and reconstructors whose numerical quadrature remains valid across discretizations (Kovachki et al., 2024, Hauck et al., 9 Jul 2025). This suggests that “resolution-invariant” performance is partly an architectural property and partly a numerical analysis condition.

4. Approximation theory and universality

Universal approximation results for EAR architectures are now available in several forms. For encoder–decoder networks on separable Banach spaces with the approximation property, there exist encoder and decoder maps and a finite-dimensional latent approximator such that every continuous operator on a compact set can be approximated arbitrarily well (Kovachki et al., 2024). DeepONet has universal approximation theorems for continuous operators $$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$6, and integral-kernel neural operators, including FNO, are universal on compact subsets because they can emulate averaging neural operators and general kernels (Kovachki et al., 2024).

The error decomposition

$$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$7

is not merely heuristic. It is the organizing principle behind sharper theorems for encoder–decoder models. In the variation-space framework, one considers operators in an infinite-dimensional structural class

$$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$8

and proves Bochner-$$N_\theta(u):=(r_\theta\circ A_\theta\circ e_\theta)(u),$$9 approximation bounds for encoder–decoder two-layer networks in which the total error decomposes into input encoding error, output encoding error, and a finite-width term of order $X$0, with a constant independent of the input and output encoding dimensions (Yang et al., 31 May 2026). When the input and output encoding errors decay polynomially with the encoding dimensions, these bounds yield algebraic approximation and learning rates (Yang et al., 31 May 2026).

A distinct topological advance concerns compact-set-independent approximation. Earlier universal theorems often permitted the approximating sequence to depend on the compact set $X$1. The EDAP framework introduces encoder–decoder approximation properties $X$2 and $X$3 that approximate the identity uniformly on every compact subset. Under EDAP and any universal finite-dimensional approximator family, every $X$4 admits a single sequence

$X$5

that converges uniformly on every compact subset of $X$6 (Gödeke et al., 31 Mar 2025). The same work shows that compact-set-independent approximation is a strictly stronger property in most relevant operator-learning settings (Gödeke et al., 31 Mar 2025).

Universality, however, does not settle efficiency. A plausible implication, reinforced by later lower-bound results, is that approximation theorems should be read as existence statements whose practical meaning depends on structural assumptions beyond continuity.

5. Statistical theory, generalization, and lower bounds

The statistical theory of EAR models separates approximation, statistical estimation, truncation, and discretization errors (Subedi et al., 4 Apr 2025). For general regression classes $X$7, least-squares-type empirical risk minimizers admit high-probability and mean-squared-error bounds in terms of localized metric entropy (Reinhardt et al., 2024). In the specific FrameNet architecture—an encoder–decoder construction based on stable frames in $X$8 and $X$9, a bounded-weight finite neural approximator on frame coefficients, and synthesis back into $Y$0—one obtains algebraic sample-complexity rates for holomorphic target operators, thereby overcoming the curse of dimensionality in those structured settings (Reinhardt et al., 2024).

These positive results coexist with strong negative ones. For very broad Lipschitz classes, lower bounds using nonlinear $Y$1-widths already indicate curse-of-dimensionality phenomena unless additional structure such as holomorphy, low intrinsic dimension, or sparse spectral content is present (Kovachki et al., 2024). The information-theoretic refinement is sharper: for the class of Lipschitz operators, metric entropy bounds imply that any bit-encoded neural operator architecture attaining accuracy $Y$2 must have size exponentially large in $Y$3, where size is measured by the number of encoded bits required to store the model in computational memory (Lanthaler, 2024). The lower bounds are proved both for uniform approximation on compact sets and for approximation in expectation under input measures with prescribed spectral decay (Lanthaler, 2024).

This is the main theoretical caution surrounding EAR. Universal approximation is compatible with severe parametric inefficiency on worst-case operator classes. Conversely, algebraic rates emerge when one restricts attention to structured subclasses, such as holomorphic PDE solution maps, operator variation spaces, low-rank latent regimes, or basis systems aligned with the input and output distributions (Reinhardt et al., 2024, Yang et al., 31 May 2026). The resulting picture is not contradictory: EAR is expressive in general, but efficient only when the operator class itself has low effective complexity.

6. PDEs, inverse problems, and other application domains

The classical application domain is PDE operator learning. For elliptic maps such as Darcy flow, the encoder augments coefficients with coordinates and geometry, the approximator captures Green’s-function-like nonlocal coupling, and the reconstructor projects latent channels to the solution while optionally enforcing boundary constraints (Kovachki et al., 2024). For time-dependent transport, Burgers, and Navier–Stokes operators, encoders include initial data, parameters, and time; approximators capture long-range propagation and spectral interactions; reconstructors may incorporate periodicity, conserved quantities, divergence-free projections, or pressure recovery (Kovachki et al., 2024).

Structure preservation is most explicit in SPON. There, finite-element spaces $Y$4 and $Y$5 define the encoder and decoder through FE projection and FE synthesis, while a graph neural processor approximates the discrete operator in the latent space. Because reconstruction stays inside the FE trial space, strong Dirichlet conditions can be enforced exactly, mixed finite-element compatibility can be inherited, and complex geometries are handled natively through the mesh and basis functions (Bouziani et al., 2024). The multigrid SPON variant couples restriction, prolongation, and interpolation operators with message passing to shift global computation to coarse levels while preserving a multi-resolution EAR structure (Bouziani et al., 2024).

EAR also extends beyond standard PDE surrogates. In opPINN for the homogeneous Fokker–Planck–Landau equation, operator surrogates first learn the diffusion and drift maps $Y$6 and $Y$7; these pretrained approximators are then embedded in a PINN whose residual enforces

$Y$8

The paper proves that the learned solution converges to the classical solution as the total loss and surrogate errors vanish (Lee et al., 2022). In limited-view photoacoustic tomography, the encoder projects observable boundary data onto a subspace spanned by training waveforms, the approximator learns a linear extension from the observed to the unobserved boundary arc, and reconstruction uses an exact full-view backprojection formula. The extension error yields a corresponding reconstruction error bound (Dreier et al., 2017). In spatiotemporal graphs, SPIN realizes EAR on the graph–time product space, using masked attention aligned with observed entries to define a stable, non-autoregressive imputation operator (Marisca et al., 2022).

These examples show that EAR is not restricted to “solution operators” in the narrow PDE sense. It encompasses data extension operators, imputation operators on discrete graph-time domains, RKHS surrogates for controlled epidemic trajectories, and FE-based structure-preserving discretizations (Dreier et al., 2017, Marisca et al., 2022, Ziarelli et al., 2024, Bouziani et al., 2024). A plausible synthesis is that EAR is best regarded as a design grammar: encode the information-bearing part of the input, approximate the operator where structure is most tractable, and reconstruct in a space that preserves the constraints, geometry, and evaluation format required by the application.