Physics-Informed Neural Networks for Source Inversion and Parameters Estimation in Atmospheric Dispersion

TL;DR

This work addresses atmospheric dispersion by jointly inverting emission sources and estimating unknown ADE parameters using physics-informed neural networks (PINNs). It introduces a neural tangent kernel (NTK)–based adaptive loss weighting to stabilize training when data are scarce and noisy, and extends NTK theory to a source-inversion setting with multiple networks. The authors derive the NTK for the two-network case (u and f), develop a practical training algorithm, and demonstrate robust performance in 2D and 3D advection-diffusion problems with constant and non-constant coefficients. The results show accurate recovery of sources and key parameters under limited measurements, with diffusion-parameter estimation remaining more challenging and highlighting avenues for uncertainty quantification in future work.

Abstract

Recent studies have shown the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). In the fields of atmospheric science and environmental monitoring, estimating emission source locations is a central task that further relies on multiple model parameters that dictate velocity profiles and diffusion parameters. Estimating these parameters at the same time as emission sources from scarce data is a difficult task. In this work, we achieve this by leveraging the flexibility and generality of PINNs. We use a weighted adaptive method based on the neural tangent kernels to solve a source inversion problem with parameter estimation on the 2D and 3D advection-diffusion equations with unknown velocity and diffusion coefficients that may vary in space and time. Our proposed weighted adaptive method is presented as an extension of PINNs for forward PDE problems to a highly ill-posed source inversion and parameter estimation problem. The key idea behind our methodology is to attempt the joint recovery of the solution, the sources along with the unknown parameters, thereby using the underlying partial differential equation as a constraint that couples multiple unknown functional parameters, leading to more efficient use of the limited information in the measurements. We present various numerical experiments, using different types of measurements that model practical engineering systems, to show that our proposed method is indeed successful and robust to additional noise in the measurements.

Figures (18)

• Figure 1: Contour Plots of the FEM solution at $t=1$(left), Exact source function with 4 measurements locations(middle) and 15 measurement locations (right), with the the selected measurements locations represented by the triangles.
• Figure 2: Prediction and absolute error of $u$ at $t=1$ and $f$, with $1\%$ of noise.
• Figure 3: Eigenvalues in a source inversion problem of $\mathbf{K}_{bb}$, $\mathbf{K}_{zz}$, $\mathbf{K}_{rr}$ vs their index, with 15 observations on $u$.
• Figure 4: Eigenvalues in a forward context of $\mathbf{K}_{i}$, $\mathbf{K}_{u_{x}}$, $\mathbf{K}_{u_{y}}$, $\mathbf{K}_{zz}$,$\mathbf{K}_{rr}$ vs their index with $n$ representing the number of iterations.
• Figure 5: Prediction and error of $u$ and $f$ with $1\%$ of noise and average observation operator over $\delta t=30$.

Theorems & Definitions (2)

• Lemma 1
• Proof 1