A Dynamical Neural Galerkin Scheme for Filtering Problems

TL;DR

We address filtering for PDE-driven dynamics with unknown parameters by reconstructing $u(t,x)$ from partial observations using a neural surrogate with time-varying weights $\theta(t)$. The approach builds on the Neural Galerkin Scheme for forward problems, yielding a projected residual that leads to $M(\theta)\dot\theta=F(t,\theta)$ and thus a dynamical parametric approximation. The paper introduces a two-step dynamical filtering algorithm that jointly estimates the state and PDE parameters and analyzes the impact of sensor placement on reconstruction quality, demonstrated on a 1D KdV equation with $\xi^\dagger=6$. Results show that reconstruction quality strongly depends on observation locations and that moving sensors along the evolving support can substantially extend accurate inference in transport-dominated regimes.

Abstract

This paper considers the filtering problem which consists in reconstructing the state of a dynamical system with partial observations coming from sensor measurements, and the knowledge that the dynamics are governed by a physical PDE model with unknown parameters. We present a filtering algorithm where the reconstruction of the dynamics is done with neural network approximations whose weights are dynamically updated using observational data. In addition to the estimate of the state, we also obtain time-dependent parameter estimations of the PDE parameters governing the observed evolution. We illustrate the behavior of the method in a one-dimensional KdV equation involving the transport of solutions with local support. Our numerical investigation reveals the importance of the location and number of the observations. In particular, it suggests to consider dynamical sensor placement.

Figures (10)

• Figure 1: Left: Initial condition $u_0$ and its neural network approximation $\mathrm{U}(\cdot, \theta_0)$. Right: Solution to the KdV equation for $\xi^\dagger=6$. This is the solution that we seek to approximate with the filtering algorithm.
• Figure 2: Percentage of eigenvalues of $\tilde{M}_k$ greater than $10^{-6}$ in absolute value, at each time $t\in[0,4]$.
• Figure 3: Left: Reconstructed solution using $m=100$ sensors uniformly spread over $\mathcal{X}$. Right: Relative error err$_k$ on the approximation of $\xi^\dagger$.
• Figure 4: Reconstructed solution using $m=100$ sensors uniformly spread over $\mathcal{X}$ for different times.
• Figure 5: Left: Reconstructed solution using $m=10$ sensors uniformly spread over $\mathcal{X}$. Right: Relative error err$_k$ on the approximation of $\xi^\dagger$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4