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Physics-informed Blind Reconstruction of Dense Fields from Sparse Measurements using Neural Networks with a Differentiable Simulator

Ofek Aloni, Barak Fishbain

TL;DR

PhysBR addresses the challenge of reconstructing dense fields from sparse measurements by integrating a differentiable PDE solver into a neural network training loop, enabling learning from sparse pairs $(s(t), s(t+\Delta t))$ without full-field ground truth. The method enforces physical constraints through the solver $\mathcal{P}$ and a smoothness penalty, yielding reconstructions that respect boundary conditions and flow structures across Burgers' equation, Wake flow, and Kolmogorov flow. Compared to CNN baselines and classical interpolation methods, PhysBR shows superior consistency and lower mean relative $L^2$ errors, with competitive performance to RBF in some regimes, at the cost of higher computation due to differentiable simulation. This physics-informed, data-efficient approach advances dense-field reconstruction in fluids, with implications for environmental monitoring and engineering in data-scarce settings, while highlighting the need for accurate governing equations and careful handling of chaotic dynamics.

Abstract

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art methods either use spatial statistics or rely on examples of dense fields in the training phase, which often are not available, and thus rely on synthetic data. Here, we present a reconstruction method that generates dense fields from sparse measurements, without assuming availability of the spatial statistics, nor of examples of the dense fields. This is made possible through the introduction of an automatically differentiable numerical simulator into the training phase of the method. The method is shown to have superior results over statistical and neural network based methods on a set of three standard problems from fluid mechanics.

Physics-informed Blind Reconstruction of Dense Fields from Sparse Measurements using Neural Networks with a Differentiable Simulator

TL;DR

PhysBR addresses the challenge of reconstructing dense fields from sparse measurements by integrating a differentiable PDE solver into a neural network training loop, enabling learning from sparse pairs without full-field ground truth. The method enforces physical constraints through the solver and a smoothness penalty, yielding reconstructions that respect boundary conditions and flow structures across Burgers' equation, Wake flow, and Kolmogorov flow. Compared to CNN baselines and classical interpolation methods, PhysBR shows superior consistency and lower mean relative errors, with competitive performance to RBF in some regimes, at the cost of higher computation due to differentiable simulation. This physics-informed, data-efficient approach advances dense-field reconstruction in fluids, with implications for environmental monitoring and engineering in data-scarce settings, while highlighting the need for accurate governing equations and careful handling of chaotic dynamics.

Abstract

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art methods either use spatial statistics or rely on examples of dense fields in the training phase, which often are not available, and thus rely on synthetic data. Here, we present a reconstruction method that generates dense fields from sparse measurements, without assuming availability of the spatial statistics, nor of examples of the dense fields. This is made possible through the introduction of an automatically differentiable numerical simulator into the training phase of the method. The method is shown to have superior results over statistical and neural network based methods on a set of three standard problems from fluid mechanics.
Paper Structure (13 sections, 18 equations, 4 figures, 1 table)

This paper contains 13 sections, 18 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Problem Formulation
  4. Construction of the Reconstructor
  5. Case Studies
  6. Burgers' Equation
  7. Wake Flow (Bounded Navier-Stokes with Obstacle)
  8. Kolmogorov Flow (Unbounded Navier-Stokes with Periodic Forcing)
  9. Comparison Methods
  10. Training Procedure
  11. Results
  12. Discussion
  13. Conclusion

Figures (4)

  • Figure 1: PhysBR training loop - schematic displaying the pipeline for learning the weights in $f_\theta$. First, an initial dense field is generated from the sparse samples $s(t)$ using the neural network $f_\theta$. Then, it is advanced in time using the differentiable numerical solver $\mathcal{P}$. Finally, it is sampled by multiplication with $\mathds{1}_S$, and loss is compared relative to the ground truth field values at sample locations $s(t+\Delta t)$.
  • Figure 2: 1d Burgers equation - reconstruction of a single sample $s(t)$ from the test set by the each method, compared to the ground truth $u(t+\Delta t,x)$.
  • Figure 3: Wake flow (Incompressible flow with obstacle) - visual comparison between the vorticity $\omega=\nabla\times\mathbf{u}$ of reconstructions of $\mathbf{u}(t+\Delta t,x,y)$ using different methods, compared to the ground truth. Vorticity is shown for visualization only. Heatmap range is fixed in all subplots according to the ground truth.
  • Figure 4: Visual comparison of Kolmogorov Flow reconstruction. The left panel shows the ground truth vorticity field. The table compares PhysBR and the other four reconstruction methods. In the top row there are $|S|=3,200$ sample locations, and on the bottom row $|S|=256$. Heatmap range is fixed in all subplots according to the ground truth.