Deep Learning for Robotic Mass Transport Cloaking

Abstract

We consider the problem of Mass Transport Cloaking using mobile robots. The robots move along a predefined curve that encloses the safe zone and carry sources that collectively counteract a chemical agent released in the environment. The goal is to steer the mass flux around a desired region so that it remains unaffected by the external concentration. We formulate the problem of controlling the robot positions and release rates as a PDE-constrained optimization, where the propagation of the chemical is modeled by the Advection-Diffusion (AD) PDE. We use a Deep Neural Network (NN) to approximate the solution of the PDE. Particularly, we propose a novel loss function for the NN that utilizes the variational form of the AD-PDE and allows us to reformulate the planning problem as an unsupervised modelbased learning problem. Our loss function is discretization-free and highly parallelizable. Unlike passive cloaking methods that use metamaterials to steer the mass flux, our method is the first to use mobile robots to actively control the concentration levels and create safe zones independent of environmental conditions. We demonstrate the performance of our method in simulations.

VarNet: Variational Neural Networks for the Solution of Partial Differential Equations

“…Using NNs to approximate solutions of PDEs can be beneficial for the following reasons: (i) their evaluation is extremely fast and thus, unlike currently available MOR methods, there is no need to compromise accuracy for speed, (ii) parallelization of the training is trivial, and (iii) the resulting model is smooth and differentiable and thus, it can be readily used in PDE-constrained optimization problems, e.g., for source identification [2] or control of PDEs [3]. The authors in [4,Ch. 04] provide a review of different approaches for solving PDEs using NNs.…”

“…The proposed loss function was first introduced and used to solve a robotic PDE-constrained optimal control problem in our short paper [3]. Compared to [3], here we introduce the VARNET library that employs the same loss function but is additionally equipped with the proposed adaptive method to optimally select the NN training points and can be used not only to solve PDEs but also for MOR.…”

“…The proposed loss function was first introduced and used to solve a robotic PDE-constrained optimal control problem in our short paper [3]. Compared to [3], here we introduce the VARNET library that employs the same loss function but is additionally equipped with the proposed adaptive method to optimally select the NN training points and can be used not only to solve PDEs but also for MOR. Finally, compared to [3], this paper presents extensive numerical experiments to validate the VARNET algorithm.…”

“…As discussed in Section I-A, Problem II.2 can be solved in different ways. The simplest approach is to use supervised learning to obtain an approximate solution of the PDE using labeled data collected using numerical methods and a loss function defined by objective (3). In this approach, the NN acts merely as a memory cell and an interpolator to store the solution more efficiently than storing the values at all grid points.…”

AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems

“…For parametric problems, we take parameters as the additional inputs of neural networks. This approach is used to solve parametric forward problems [24] and control problems [49]. A typical way for sampling training points is to separately sample data in Ω and P to get {x i } and {µ j }, and then compose product data {(x i , µ j )} in Ω × P. Rather than taken from each slice of Ω(µ) for a fixed µ, collocation points are sampled in space Ω P in this work, where Ω P = {x(µ) : x ∈ Ω(µ)} represents the joint spatio-parametric domain.…”