Physics-Informed Multiresolution Wavelet NN

• The paper demonstrates that employing a fixed multiresolution wavelet basis in a physics-informed framework enhances accuracy and speed for solving PDEs.
• It uses a least-squares formulation on collocation points to enforce PDE and boundary conditions, sidestepping the nonconvex challenges of conventional PINNs.
• The approach effectively handles both stationary and time-dependent PDEs, especially for high-frequency, oscillatory, or localized phenomena.

Physics-Informed Multiresolution Wavelet Neural Network (PIMWNN) denotes a class of PDE solvers in which the unknown solution is approximated by a multiresolution wavelet neural network, substituted directly into the governing equation, and fitted through physics-based residual equations and least squares rather than conventional end-to-end nonlinear PINN optimization. In the formulation introduced in “Physics-informed Multiresolution Wavelet Neural Network Method for Solving Partial Differential Equations” (Han et al., 11 Aug 2025), the method combines the collocation-based, physics-informed idea of PINNs with a fixed multiresolution wavelet basis, and is presented as a mesh-free approach for stationary and time-dependent PDEs. Its stated motivation is to improve accuracy and speed relative to standard PINNs and to address spectral bias, especially in regimes with high-frequency, oscillatory, or localized structure; this motivation is consistent with reports that standard PINNs become difficult to train on multi-scale propagating wavefields and on sharp boundary layers (Moseley et al., 2020, Nayak et al., 8 Jun 2026).

1. Definition and conceptual scope

PIMWNN is built from two ingredients. The first is the physics-informed training paradigm: interior PDE residuals and boundary or initial conditions are enforced at collocation points. The second is the multiresolution wavelet neural network (MWNN) ansatz: the solution is represented by scaling functions and wavelets across several resolution levels. In the 2025 formulation, the central workflow is explicit: use the MWNN to approximate unknown functions, substitute the MWNN into PDEs, evaluate the resulting equations at collocation points, and determine the unknown output-layer coefficients by least squares (Han et al., 11 Aug 2025).

This construction distinguishes PIMWNN from the standard PINN template in two ways. First, the approximation space is not a generic coordinate-to-solution multilayer perceptron; it is an interpretable wavelet expansion whose refinement is controlled by the finest scale $J$. Second, training is not described as a nonconvex gradient-based search over all hidden weights and biases. The basis functions are fixed, and the trainable parameters are the outer-layer coefficients only. A plausible implication is that PIMWNN inherits some of the approximation advantages of wavelets—localization in space and explicit scale separation—while also avoiding part of the optimization burden that characterizes standard PINNs.

The method is presented for both stationary and time-dependent PDEs. For stationary problems,

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$

and for time-dependent problems,

$$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$

The derivation and experiments explicitly formulate the Dirichlet case, although the method is described more generally as able to handle Dirichlet, Neumann, and Robin boundary conditions (Han et al., 11 Aug 2025).

2. Multiresolution wavelet approximation

The multiresolution structure follows a standard MRA. The method uses nested subspaces

$$V_j \subset V_{j+1} \subset \cdots \subset L^2(\mathbb{R}), \quad j\in \mathbb{Z},$$

with

$$\bigcap V_j = \{0\}, \qquad \overline{\bigcup V_j}=L^2,$$

a scaling function $\phi$, and wavelet spaces $W_j$ satisfying

$V_j \oplus W_j = V_{j+1}.$

The associated decomposition is

$$f(t) = \sum_{k\in\mathbb{Z}} \langle f,\phi_{0,k}\rangle \phi_{0,k}(t) + \sum_{j\ge 0}\sum_{k\in\mathbb{Z}} \langle f,\psi_{j,k}\rangle \psi_{j,k}(t).$$

In the concrete implementation, the paper uses Shannon wavelets on $[0,L]$, with

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$0

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$1

and

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$2

The 1D multiresolution approximation is

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$3

which the network rewrites as

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$4

For PDE solving, the unknown is represented as

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$5

or, in space-time,

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$6

For two variables, tensor-product bases are used. The 2D scaling basis is

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$7

and the 2D wavelets are

$$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$8

Thus the hidden layer is not generic: it is partitioned into a coarse scaling subnetwork and wavelet subnetworks at successive resolutions. Increasing $$\mathcal{A}u = f, \quad x\in \Omega, \qquad \mathcal{B}u = g, \quad x\in \Gamma,$$9 adds finer basis elements rather than merely widening an MLP (Han et al., 11 Aug 2025).

3. Physics-informed least-squares formulation

For stationary PDEs, substituting the MWNN ansatz into the PDE and Dirichlet boundary conditions gives

$$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$0

$$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$1

With interior collocation points $$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$2 and boundary collocation points $$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$3, the method assembles

$$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$4

where $$\mathcal{A}u = f, \quad x\in \Omega \times I, \qquad \mathcal{B}u = g, \quad x\in \Gamma \times I, \qquad \mathcal{C}u = h, \quad x\in \Omega \times \{0\}.$$5, (B_k=\