---
kind: concept
slug: wavelet-transform
title: Wavelet transform
summary: >-
  Splits a signal into coarse scaling coefficients and detail (wavelet) coefficients at each scale, and rebuilds it from them. One card so far: a multiwavelet decomposition with fixed Legendre or Chebyshev filters, with small learned kernels per scale inside a PDE operator. Author-reported errors are lower than FNO's on most PDE tables here; the Chebyshev variant is weaker than Legendre.
tags: [wavelets, multiresolution, neural-operator, pde]
papers: ["2109.13459"]
related: [neural-operator, fourier-neural-operator]
status: draft
updated_at: 2026-10-18
---

## Overview

A wavelet transform writes a function as a coarse summary plus detail terms at
each scale. The variant on file is a multiwavelet basis: piecewise polynomials
of degree below k on dyadic intervals form nested spaces V_n, and each level adds
a wavelet space W_n orthogonal to V_n [arxiv:2109.13459 §2.2] [eq. 2] [eq. 3].

The filter matrices H and G map the finer scaling coefficients to the next
coarser scaling coefficients s and detail coefficients d [eq. 6] [eq. 7]. The
inverse uses the same filters, corrected by Gram matrices Sigma, which are the
identity for the Legendre basis [eq. 8] [eq. 9] [app. C.1]. In 2D and 3D the
filters are Kronecker products of the 1D ones [arxiv:2109.13459 §2.2].

The paper's premise is that the kernel of a smooth integral operator, apart from
finitely many singularities, is sparse in this basis [arxiv:2109.13459 §2.1]
[§2.4]. The model then learns a small kernel for each scale
[arxiv:2109.13459 §2.3].

The wiki holds one paper for this concept, so this page describes the transform
as that paper uses it.

## Variants

### Legendre multiwavelets

The polynomials are orthogonal with respect to the uniform measure on [0, 1]
[arxiv:2109.13459 app. C.2]. Sigma is the identity [app. C.1], so the
reconstruction filters equal the decomposition ones. The paper's headline model
uses this basis [arxiv:2109.13459 §3]. Its reported error is the lowest of the
listed baselines on KdV, Burgers, Darcy and Navier-Stokes [tab. 1] [tab. 2]
[tab. 3] [tab. 7]. Card: [Multiwavelet-based Operator Learning for Differential Equations](/papers/2109.13459).

### Chebyshev multiwavelets

The polynomials are orthogonal with respect to the weight
1/sqrt(1 - (2x-1)^2) on [0, 1] [arxiv:2109.13459 app. C.3]. The paper's eq. 8
and eq. 9 put Sigma on the reconstruction side only [eq. 8] [eq. 9]. The paper
extends multiwavelets to non-uniform measures, which is what allows this basis
[arxiv:2109.13459 §2.2] [app. B]. In the card's tables the Chebyshev model has a
higher error than Legendre at every resolution listed [tab. 1] [tab. 2]
[tab. 7]. Against FNO it is mixed: lower error on the KdV and Darcy tables, but
higher on Burgers at s = 256 [tab. 7] and on Navier-Stokes at nu = 1e-5
[tab. 3].

### Random filters (control)

The paper's control, MWT Rnd, uses random filters. Its Darcy error is about 0.24
at every resolution [tab. 2], far above the fixed filters. The paper's claim
that random filters do not learn is not checked by the authors' code, which
does not implement this row (see the card).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Smooth PDE data, and the grid may change between training and use | Legendre multiwavelet | Lower error than FNO on the KdV and Darcy tables [tab. 1] [tab. 2]. The same small kernels act at every scale [arxiv:2109.13459 §2.3]. |
| Test resolution far above training resolution | Check first | For a model trained at s = 512, error rises from 0.0140 at s = 2048 to 0.0241 at s = 8192 on the paper's Burgers rows [tab. 4]. |
| The integration measure must be non-uniform | Chebyshev multiwavelet | It is the basis that the paper's extension to non-uniform measures allows [arxiv:2109.13459 §2.2] [app. B]. Expect a higher error than Legendre [tab. 1] [tab. 7]. |
| Simpler implementation, no Sigma | Legendre multiwavelet | Sigma is the identity [app. C.1]. Chebyshev needs Sigma [eq. 8] [eq. 9], and the paper gives two node counts for its quadrature [app. C.3] [app. C.4]. |
| Choosing the filter order k | k in [1, 6] | The paper recommends this range and reports drift from orthogonality above k = 20 in floating point [app. C.4]. The paper's own experiments use k = 4 and k = 3 [§3] [app. D.1]. |
| Grid size is not a power of two | Test before use | The paper's resolution claim is tied to grids of this kind, and the card's implementation notes flag a limit here (see the card). |

The paper's Legendre headline model uses 1000 training and 200 test samples
[§3].

## Implementation sketch

Language-agnostic core of one decomposition and one reconstruction step. The
filters H0, H1, G0, G1 come from the Legendre or Chebyshev construction
[app. C.2] [app. C.3]. The learned operator acts between the two steps [eq. 11].

```pseudocode
# One decomposition step, fine to coarse  [eq. 6] [eq. 7]
# Input:  s_fine (2M, k) scaling coefficients on the finer grid
# Output: s (M, k) coarse scaling, d (M, k) wavelet coefficients

def decompose(s_fine):
    # Even and odd fine nodes, paired as in eq. 6 and eq. 7
    s_even = s_fine[0::2]
    s_odd = s_fine[1::2]
    # Coarse scaling coefficients from both halves  [eq. 6]
    s = s_even @ H0.T + s_odd @ H1.T
    # Wavelet (detail) coefficients  [eq. 7]
    d = s_even @ G0.T + s_odd @ G1.T
    return s, d
```

```pseudocode
# One reconstruction step, coarse to fine  [eq. 8] [eq. 9]
# Input:  s (M, k), d (M, k)
# Output: s_fine (2M, k)

def reconstruct(s, d):
    # Even and odd fine nodes, each corrected by its Sigma  [eq. 8] [eq. 9]
    s_even = (s @ H0 + d @ G0) @ Sigma0.T
    s_odd = (s @ H1 + d @ G1) @ Sigma1.T
    # Interleave the two sets back onto the fine grid
    return interleave(s_even, s_odd)
```

Implementation notes:

- The paper's eq. 7 is printed with H^(1) in the second term. The matrix form in
  the appendix uses G^(1) [app. C.1], and the sketch above uses G1. Use G1
  [arxiv:2109.13459 §2.2] [app. C.1].
- Between the two steps, a learned operator acts on d and s: U_d = A d + B s,
  U_s = C d, with a learned coarsest-scale map [eq. 11]. The paper's operator
  is written this way, and the paper reuses A, B and C at every scale
  [arxiv:2109.13459 §2.3].
- For Chebyshev, keep the decomposition on the raw H and G, and apply Sigma only
  in reconstruction [eq. 8] [eq. 9]. Compute Sigma and G with the quadrature
  rather than copying the four-decimal values in the appendix [app. C.3].

## Contradictions

- **Filter order for Navier-Stokes.** The paper states k = 4 for all experiments
  [§3]. Its Navier-Stokes appendix uses k = 3 [app. D.1].
- **Kernel type, paper and code.** The paper describes A, B and C as
  single-layer CNNs following a linear layer [§2.3] [§3], and for Navier-Stokes
  a single-layer 3D convolution [app. D.1]. The authors' 1D code uses
  Fourier-domain kernels for all three [code: models/models.py#L101-L103]. The
  2D and 3D code uses spectral kernels for A and convolutions for B and C
  [code: models/models.py#L318-L320]. Implement the code's version if you want
  to match its numbers, and say so.
- **Headline numbers.** The abstract quotes 0.0033 for KdV and 0.0018 for
  Burgers. The tables give 0.00338 to 0.00418 for KdV [tab. 1] and 0.00178 to
  0.00199 for Burgers [tab. 7].
- **Burgers at s = 2048.** Section 3.5 gives 0.00189, and the table gives 0.00186
  [§3.5] [tab. 7].

## Papers

- [Multiwavelet-based Operator Learning for Differential Equations](/papers/2109.13459): a multiwavelet decomposition with Legendre or Chebyshev filters and learned kernels per scale, compared with FNO and other neural operators on PDE benchmarks.