Neural Inverse Source Problems

TL;DR

This work proposes a Physics-Informed Neural Network based approach for solving the inverse source problems in robotics, jointly identifying unknown source functions and the complete state of a system given partial and noisy observations.

Abstract

Reconstructing unknown external source functions is an important perception capability for a large range of robotics domains including manipulation, aerial, and underwater robotics. In this work, we propose a Physics-Informed Neural Network (PINN [1]) based approach for solving the inverse source problems in robotics, jointly identifying unknown source functions and the complete state of a system given partial and noisy observations. Our approach demonstrates several advantages over prior works (Finite Element Methods (FEM) and data-driven approaches): it offers flexibility in integrating diverse constraints and boundary conditions; eliminates the need for complex discretizations (e.g., meshing); easily accommodates gradients from real measurements; and does not limit performance based on the diversity and quality of training data. We validate our method across three simulation and real-world scenarios involving up to 4th order partial differential equations (PDEs), constraints such as Signorini and Dirichlet, and various regression losses including Chamfer distance and L2 norm.

Figures (14)

• Figure 1: Modified MLP with $L$ layers. Green arrows indicate fully connected layers, and red arrows indicate operations between the fully connected layer output, $\bm{a}$, and $\bm{b}$. $L$ is the number of the green box module (Eq. \ref{['eq: green box']}) from the input $x$ to the output $\Psi_{\theta}(\bm{x})=(\bm{y}(\bm{x}), \bm{f}(\bm{x}))$.
• Figure 2: Real-world data collection setup using Soft Bubbles with 6 designed objects.
• Figure 3: The left panel shows our double pendulum system with positive displacement to the right and positive angles in the counterclockwise direction. In the right panel, we include results without noise and with noise $\sim \mathcal{N}(0,0.01)$ and $\sim \mathcal{N}(0,0.03)$ in observations. The scatter plots in the first row depict $q_m$ (red), $\psi_m$ (blue), and $\phi_m$ (green), overlaid with the estimated states $q_\theta$ (orange), $\psi_\theta$ (blue), and $\phi_\theta$ (green) represented by lines. The second row shows the estimated force $f_\theta$ (orange) and the ground truth $f_m$ (blue).
• Figure 4: A) Partial pointcloud measurement (green), B) reconstructed deformed geometry where the color represents predicted contact pressure ranging from 0 to 4,000Pa, C) reconstructed deformed geometry overlaid with ground truth contact (blue) and partial observation (green), D) estimated contact location (red) overlaid with the ground truth (blue), E) baseline's contact location (yellow) overlaid with the ground truth (blue).
• Figure 5: A) Visualization of real-world Soft Bubble and object (pink) interactions. The Soft Bubble is from our model's reconstruction, and the color indicates our model's contact pressure predictions. B) Real-world Soft Bubble point cloud measurement with holes and occlusions. The color represents heights from the x-y plane. The dotted line indicates the actual bubble dimension. C) Ground truth contact mask prediction (navy). D) Estimated contact pressure overlaid with the ground truth contact location indicated in outlines (blue).