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An Efficient Wavelet-based Physics Informed Residual Neural Networks for Flow Field Reconstruction with Extremely Sparse Data

Biswanath Barman, Rajendra K. Ray

TL;DR

This work tackles the challenge of reconstructing high-fidelity flow fields from extremely sparse velocity measurements. It introduces Wavelet-Physics-Informed Residual Neural Networks (W-PIRNNs) that fuse a learnable wavelet activation $W(t) = w_1 \sin(t) + w_2 \cos(t)$ with residual connections to enforce Navier-Stokes physics while predicting velocity, pressure, vorticity, and streamlines. The approach achieves accurate reconstructions for cylinder wake flows at $\text{Re}=100$ using as little as $0.05\%$ supervision and extends to forward/inverse problems including Burgers and Schrödinger equations, demonstrating parameter identification accuracy and periodic boundary handling. Compared with standard PINNs and FFPICN, W-PIRNNs converge faster (around 2000 epochs) and require far less data, enabling robust, physics-consistent flow reconstruction under extreme data scarcity with significant practical impact for experimental fluid mechanics and data assimilation.

Abstract

This paper introduces wavelet-physics-informed residual neural networks (W-PIRNNs) to study complex fluid flow problems by reconstructing the flow field from highly sparse, supervised data. Our W-PIRNNs fundamentally integrate ResNet and employ the wavelet $W(t) = w_1 \sin(t) + w_2 \cos(t)$ as an activation function. Due to the vanishing and ballooning gradient problems associated with typical PINNs' deep networks, we implemented residual-based skip connections. Our W-PIRNNs, which integrate supervised data with physical principles, demonstrate efficacy even in scenarios of sparse or partial data, enabling the reconstruction of flow fields using merely $0.05\%$ velocity data for training. The wake flow around a circular cylinder served as the test case for our proposed technique, which depends exclusively on velocity data for training. This technique facilitates the precise reconstruction of velocity, pressure, streamlines, and vorticity, requiring fewer epochs and less processing time. Significantly, our proposed W-PIRNNs effectively resolve PDEs in both forward and inverse contexts. Burger's equation served as a test case for both the forward and inverse problem configurations. Our network calculates the diffusion or viscosity coefficient ($λ_2$) with an absolute error of $0.065\%$ and the convection coefficient ($λ_1$) with an absolute error of $0.002\%$. Furthermore, the Schrödinger equation is examined in the forward setting to assess the framework's ability to handle periodic boundary conditions. To the best of our knowledge, W-PIRNNs represent the first method capable of flow reconstruction using highly sparse supervised data, as well as reconstructing streamline and vorticity, and they effectively address both forward and inverse problems with high accuracy.

An Efficient Wavelet-based Physics Informed Residual Neural Networks for Flow Field Reconstruction with Extremely Sparse Data

TL;DR

This work tackles the challenge of reconstructing high-fidelity flow fields from extremely sparse velocity measurements. It introduces Wavelet-Physics-Informed Residual Neural Networks (W-PIRNNs) that fuse a learnable wavelet activation with residual connections to enforce Navier-Stokes physics while predicting velocity, pressure, vorticity, and streamlines. The approach achieves accurate reconstructions for cylinder wake flows at using as little as supervision and extends to forward/inverse problems including Burgers and Schrödinger equations, demonstrating parameter identification accuracy and periodic boundary handling. Compared with standard PINNs and FFPICN, W-PIRNNs converge faster (around 2000 epochs) and require far less data, enabling robust, physics-consistent flow reconstruction under extreme data scarcity with significant practical impact for experimental fluid mechanics and data assimilation.

Abstract

This paper introduces wavelet-physics-informed residual neural networks (W-PIRNNs) to study complex fluid flow problems by reconstructing the flow field from highly sparse, supervised data. Our W-PIRNNs fundamentally integrate ResNet and employ the wavelet as an activation function. Due to the vanishing and ballooning gradient problems associated with typical PINNs' deep networks, we implemented residual-based skip connections. Our W-PIRNNs, which integrate supervised data with physical principles, demonstrate efficacy even in scenarios of sparse or partial data, enabling the reconstruction of flow fields using merely velocity data for training. The wake flow around a circular cylinder served as the test case for our proposed technique, which depends exclusively on velocity data for training. This technique facilitates the precise reconstruction of velocity, pressure, streamlines, and vorticity, requiring fewer epochs and less processing time. Significantly, our proposed W-PIRNNs effectively resolve PDEs in both forward and inverse contexts. Burger's equation served as a test case for both the forward and inverse problem configurations. Our network calculates the diffusion or viscosity coefficient () with an absolute error of and the convection coefficient () with an absolute error of . Furthermore, the Schrödinger equation is examined in the forward setting to assess the framework's ability to handle periodic boundary conditions. To the best of our knowledge, W-PIRNNs represent the first method capable of flow reconstruction using highly sparse supervised data, as well as reconstructing streamline and vorticity, and they effectively address both forward and inverse problems with high accuracy.
Paper Structure (21 sections, 2 theorems, 46 equations, 33 figures, 10 tables, 1 algorithm)

This paper contains 21 sections, 2 theorems, 46 equations, 33 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Physics-informed neural networks (PINNs)
  4. Residual Neural Network (ResNet)
  5. Min-Max Normalization
  6. Wavelet activation function
  7. Proposed wavelet-physics informed residual neural networks (W-PIRNNs)
  8. Numerical experiments
  9. Numerical dataset for both training and validation at Re=100
  10. Flow reconstruction from sparse data for a circular cylinder at Re=100
  11. Loss Functions and Training Strategy:
  12. Relative $L_2$ Error Metrics:
  13. Training Duration and Observations:
  14. Reconstructing flow fields using different percentages of sparse data (e.g., 100%, 20%, 5%, and 1%)
  15. Flow field reconstruction with extremely sparse data ($0.16\%$ and $0.05\%$ supervision)
  16. ...and 6 more sections

Key Result

Theorem 1

hornik1991approximation Let $\sigma$ be a continuous, bounded, and non-constant activation function. Then for any continuous function $\mathrm{f}$ defined on a compact subset $K \subset \mathbb{R}^n$ and for any $\epsilon>0$, there exists a feedforward neural network with a single hidden layer and a In other words, this theorem states that a feedforward NN, with at least one hidden layer containin

Figures (33)

  • Figure 1: Schematic of PINNs' architecture to solve partial differential equations (PDEs).
  • Figure 2: Residual learning: a building block.
  • Figure 3: The graph of $A \cos(\omega t - \phi)$
  • Figure 4: Plots of (a) the wavelet activation function $W(t) = w_1\sin(t) + w_2\cos(t)$ and (b) its derivative $W'(t) = w_1\cos(t) - w_2\sin(t)$, evaluated at three different parameter combinations: $(w_1, w_2) = (1, 0)$, $(0, 1)$, and $(0.7, 0.5)$.
  • Figure 5: The vorticity contour of the flow past a circular cylinder with dynamic vortex shedding at Re=100.
  • ...and 28 more figures

Theorems & Definitions (2)

  • Theorem 1: Universal Approximation Theorem
  • Theorem 2