Basis-to-Basis Operator Learning

• BASIS is an operator-learning framework that transforms functions between Hilbert spaces using learned neural network bases and least-squares projections.
• It constructs finite-dimensional representations that enable efficient learning of both linear and nonlinear operators with superior accuracy.
• The method preserves structural properties such as linearity and homogeneity while adapting to irregular sampling and complex domains.

Basis-to-Basis (B2B) operator learning is an operator-learning framework for maps between Hilbert spaces of functions that treats an operator as a map between coefficient vectors in learned input and output bases. In this formulation, neural-network basis functions are learned for the input and output function spaces, coefficients are recovered from sampled function values by least squares, and the operator itself is learned in the resulting finite-dimensional coefficient space. The method is derived from function encoders, accommodates sample locations that are not fixed across functions, is especially effective for linear operators through a closed-form matrix fit, and admits variants directly analogous to eigendecomposition and singular value decomposition. On seven benchmark operator-learning tasks, it reports two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmarks (Ingebrand et al., 2024).

1. Hilbert-space formulation and basis-to-basis representation

B2B is formulated for an operator

$\mathcal{T}:\mathcal{G}\to\mathcal{H},$

where the input space is a Hilbert space of functions

$\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$

and the output space is another Hilbert space of functions

$\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$

with $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$ Euclidean. A typical example is a PDE solution operator mapping an input function $f\in\mathcal{G}$ to a solution $\mathcal{T}f\in\mathcal{H}$ (Ingebrand et al., 2024).

The data model is sample-based rather than grid-fixed. For each training instance, one observes

$$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$

and the locations $x_i$ and $y_i$ need not be fixed across $n$. This is a central departure from grid-bound operator learners (Ingebrand et al., 2024).

The basis-to-basis decomposition has three parts. First, the input function is approximated in a learned basis

$\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$0

where $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$1 are neural-network basis functions. Second, the output is approximated in a learned basis

$\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$2

where $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$3 are neural-network basis functions. Third, the operator is learned as a map between coefficient spaces,

$\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$4

either linearly through $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$5 with $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$6, or nonlinearly through a feed-forward network on coefficients (Ingebrand et al., 2024).

This representation is basis-centric rather than pointwise. The paper’s central claim is that an operator on functions is most naturally viewed as a transformation between low-dimensional coefficient descriptions of those functions, with the bases themselves learned from data (Ingebrand et al., 2024).

2. Function encoders, coefficient recovery, and sampling flexibility

The basis functions $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$7 and $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$8 are neural networks, but the coefficients are not produced by an encoder network. They are computed deterministically by least squares. For input samples $\mathcal{G}=\{f\mid f:\mathcal{X}\to\mathcal{Z}\},$9, define the design matrix $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$0 by

$\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$1

and the data vector $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$2. The coefficient vector $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$3 solves

$\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$4

with closed-form solution

$\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$5

The output coefficients $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$6 are obtained analogously from an output design matrix $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$7 (Ingebrand et al., 2024).

These are empirical normal equations induced by the sampled data. No explicit analytic inner product on the function spaces is required. The method therefore uses least-squares projection onto a linearly independent set of basis functions rather than an orthonormal basis. This directly addresses a common misconception: B2B does not require conditions such as $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$8 (Ingebrand et al., 2024).

The input function encoder is the pair consisting of the learned basis $\mathcal{H}=\{h\mid h:\mathcal{Y}\to\mathcal{W}\},$9 and the deterministic least-squares encoder $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$0. The basis parameters are trained by minimizing the reconstruction loss

$\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$1

An analogous procedure is used for the output basis $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$2. During training, gradients backpropagate through the least-squares solve into the basis parameters, so the basis is adapted to the data subspace while coefficient computation remains closed-form (Ingebrand et al., 2024).

Because the design matrices $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$3 and $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$4 are constructed from whatever sample locations are actually observed, B2B does not require fixed sensor locations. The computational bottleneck is inversion of a $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$5 or $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$6 matrix, so cost depends on basis size rather than on a fixed grid. The paper contrasts this explicitly with DeepONet’s branch network, which expects a fixed-length input vector (Ingebrand et al., 2024).

3. Finite-dimensional operator maps and preservation of linear structure

For linear operators, B2B reduces operator learning to a least-squares fit for a single matrix $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$7. Given training pairs $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$8, one first computes coefficient vectors $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{W}$9 and $f\in\mathcal{G}$0, then solves

$f\in\mathcal{G}$1

If

$f\in\mathcal{G}$2

then the standard normal-equations solution is

$f\in\mathcal{G}$3

when $f\in\mathcal{G}$4 is invertible (Ingebrand et al., 2024).

For nonlinear operators, the basis representation is retained, but the map $f\in\mathcal{G}$5 is learned by a feed-forward network in coefficient space. The resulting functional approximation is obtained by decoding the predicted $f\in\mathcal{G}$6 in the learned output basis (Ingebrand et al., 2024).

A notable structural property appears in the linear case. The paper’s Theorem 1 states that if $f\in\mathcal{G}$7 and $f\in\mathcal{G}$8 is linear, then the learned approximation satisfies

$f\in\mathcal{G}$9

The argument is straightforward: the least-squares coefficient map is linear in sampled function values, matrix multiplication by $\mathcal{T}f\in\mathcal{H}$0 is linear, and reconstruction in the output basis is linear. This yields guaranteed additivity and homogeneity in the learned coefficient representation, provided sampling is sufficient (Ingebrand et al., 2024).

This matters empirically. On derivative and anti-derivative benchmarks, the paper reports that B2B preserves linearity and homogeneity with small errors, whereas DeepONet errors increase by orders of magnitude in the same tests. The result suggests that explicitly encoding operator structure at the coefficient level is not merely a representational convenience but a mechanism for exact structural preservation within the learned subspace (Ingebrand et al., 2024).

4. Spectral variants and connections to functional analysis

The paper places B2B within the language of Hilbert-space operator theory. For a compact bounded linear operator $\mathcal{T}f\in\mathcal{H}$1, classical singular value decomposition takes the form

$\mathcal{T}f\in\mathcal{H}$2

with singular values $\mathcal{T}f\in\mathcal{H}$3, right singular functions $\mathcal{T}f\in\mathcal{H}$4, and left singular functions $\mathcal{T}f\in\mathcal{H}$5. B2B uses this as a template for a learned SVD-like construction in which neural basis functions $\mathcal{T}f\in\mathcal{H}$6 and $\mathcal{T}f\in\mathcal{H}$7 replace analytic singular functions, and least-squares coefficients replace exact inner products (Ingebrand et al., 2024).

The learned SVD operator is written as

$\mathcal{T}f\in\mathcal{H}$8

where the coefficients $\mathcal{T}f\in\mathcal{H}$9 are obtained by least squares against the learned input basis. The basis functions are not forced to be orthonormal; they are singular-like modes learned from data (Ingebrand et al., 2024).

For self-adjoint operators, the paper derives an eigendecomposition analogue. A shared neural basis $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$0 is used for both input and output, together with learned eigenvalues $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$1, yielding

$$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$2

Unlike the three-stage B2B pipeline, the SVD and ED variants are trained end-to-end using operator loss directly (Ingebrand et al., 2024).

These spectral variants are also used diagnostically. For a learned derivative operator, the paper plots decay of eigenvalues from the ED model and singular values from the SVD of the B2B matrix $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$3, observing rapid, approximately exponential decay characteristic of compact operators. This suggests that the learned basis exposes low-rank structure in a form directly comparable to classical spectral decompositions (Ingebrand et al., 2024).

A recurrent theme is that B2B imports functional-analytic structure without imposing the traditional rigidities of orthonormal spectral methods. The basis is learned, not prescribed; the projection is empirical least squares, not exact inner-product projection; yet the resulting algorithms remain close analogues of SVD and eigendecomposition (Ingebrand et al., 2024).

5. Implementation profile and empirical behavior

Each basis function $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$4, $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$5, $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$6, or $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$7 is a neural network, typically an MLP, with its own parameters. There is no fixed analytic dictionary such as Fourier or wavelet bases. In the basic B2B pipeline, training proceeds in phases: train the input basis, train the output basis, then freeze both and train the operator map in coefficient space. In the linear case, the last phase is replaced by the closed-form least-squares solution for $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$8 (Ingebrand et al., 2024).

Typical basis sizes are $$D_n=\Big(\{(x_i,f_n(x_i))\}_{i=1}^m,\;\{(y_i,(\mathcal{T}f_n)(y_i))\}_{i=1}^p\Big),$$9. The paper advises erring on the side of over-specifying the basis because inversion of $x_i$0 and $x_i$1 remains cheap at this scale. Ablations show that on low-dimensional polynomial problems, performance is insensitive once the basis size exceeds the true space dimension, whereas on higher-dimensional problems such as 2D Darcy flow, too few basis functions degrade performance (Ingebrand et al., 2024).

The implementation is designed to accommodate multiple input fields, vector-valued outputs, and irregular domains. Multiple input fields can be concatenated or handled as multiple channels into the input basis network. Vector-valued outputs are handled by taking $x_i$2 with $x_i$3. In the 2D Darcy problem on an L-shaped domain, the output mesh is irregular and differs from the regular grid used for inputs; the method simply evaluates the learned bases at the points available in each dataset (Ingebrand et al., 2024).

The empirical study spans seven tasks: anti-derivative, derivative, 1D nonlinear Darcy flow, 2D Darcy flow on an L-shaped domain, elastic plate, parameterized heat equation, and 1D Burgers’ equation. Selected mean MSE results over 10 runs are reported below (Ingebrand et al., 2024).

Task	B2B MSE	Vanilla DeepONet MSE
Anti-derivative	$x_i$4	$x_i$5
Derivative	$x_i$6	$x_i$7
1D Darcy	$x_i$8	$x_i$9
2D Darcy (L-shaped)	$y_i$0	$y_i$1
Elastic plate	$y_i$2	$y_i$3
Burgers’	$y_i$4	$y_i$5

Several patterns are emphasized. First, B2B often improves accuracy by one to two orders of magnitude, especially on simple linear operators and several PDEs. Second, in ablations with varying sensor locations, DeepONet variants fail to converge while B2B and its spectral variants maintain low MSE, because coefficient representations obtained by least squares are invariant to sampling locations. Third, the paper reports smoother, bowl-shaped basis-training loss landscapes for B2B than the sharp spikes observed in end-to-end DeepONet training, attributing this to decomposition of the task into basis learning and low-dimensional operator fitting (Ingebrand et al., 2024).

6. Interpretation, limitations, and extensions

B2B reframes operator learning as representation learning on function spaces. Instead of mapping one discretization to another, it learns finite-dimensional subspaces adapted to the operator and performs inference in those coordinates. This suggests a broader synthesis between neural operators and classical reduced-order modeling: B2B retains neural flexibility in the basis functions while using least squares, matrix fits, and spectral decompositions in the coefficient space (Ingebrand et al., 2024).

The approach also has interpretive advantages. In the anti-derivative example, the learned basis functions converge toward quadratic and cubic polynomial modes, illustrating that the basis-to-basis map can expose recognizable operator structure. For linear operators, the matrix $y_i$6 becomes a compact summary of the learned operator, and its singular spectrum can be analyzed directly (Ingebrand et al., 2024).

The limitations stated in the paper are instructive. No explicit orthogonality, multi-resolution organization, or locality is imposed on the learned basis; the basis functions are generic neural networks trained for reconstruction. The paper identifies natural extensions toward orthonormalization by Gram–Schmidt, wavelet-like multi-resolution or localized bases, and physics-informed bases. It also notes that where the input does not naturally inhabit a rich function space, as in the parameterized heat equation with a scalar-plus-parameter input, the advantage of basis learning is reduced and end-to-end DeepONet can perform slightly better (Ingebrand et al., 2024).

A common misunderstanding is that basis-based operator learning is only useful on regular grids or only for linear operators. The paper argues against both. B2B is explicitly designed for varying sample locations and irregular domains, and while it is especially potent for linear operators, the same coefficient-space architecture extends to nonlinear maps through a neural operator on coefficients. A plausible implication is that the principal contribution is not a specialized linear method, but a reorganization of operator learning around learned finite-dimensional function-space coordinates (Ingebrand et al., 2024).

In that sense, B2B is best understood as a functional-analytic reformulation of neural operator learning: learn the input and output subspaces, encode observations into coefficients by least squares, and learn the operator where its algebraic structure is most explicit.