KEEC: Embed to Control on An Equivariant Geometry

This paper investigates how representation learning can enable optimal control in unknown and complex dynamics, such as chaotic and non-linear systems, without relying on prior domain knowledge of the dynamics. The core idea is to establish an equivariant geometry that is diffeomorphic to the manifold defined by a dynamical system and to perform optimal control within this corresponding geometry, which is a non-trivial task. To address this challenge, Koopman Embed to Equivariant Control (KEEC) is introduced for model learning and control. Inspired by Lie theory, KEEC begins by learning a non-linear dynamical system defined on a manifold and embedding trajectories into a Lie group. Subsequently, KEEC formulates an equivariant value function equation in reinforcement learning on the equivariant geometry, ensuring an invariant effect as the value function on the original manifold. By deriving analytical-form optimal actions on the equivariant value function, KEEC theoretically achieves quadratic convergence for the optimal equivariant value function by leveraging the differential information on the equivariant geometry. The effectiveness of KEEC is demonstrated in challenging dynamical systems, including chaotic ones like Lorenz-63. Notably, our findings indicate that isometric and isomorphic loss functions, ensuring the compactness and smoothness of geometry, outperform loss functions without these properties.

Safe Reinforcement Learning in Tensor Reproducing Kernel Hilbert Space

This paper delves into the problem of safe reinforcement learning (RL) in a partially observable environment with the aim of achieving safe-reachability objectives. In traditional partially observable Markov decision processes (POMDP), ensuring safety typically involves estimating the belief in latent states. However, accurately estimating an optimal Bayesian filter in POMDP to infer latent states from observations in a continuous state space poses a significant challenge, largely due to the intractable likelihood. To tackle this issue, we propose a stochastic model-based approach that guarantees RL safety almost surely in the face of unknown system dynamics and partial observation environments. We leveraged the Predictive State Representation (PSR) and Reproducing Kernel Hilbert Space (RKHS) to represent future multi-step observations analytically, and the results in this context are provable. Furthermore, we derived essential operators from the kernel Bayes' rule, enabling the recursive estimation of future observations using various operators. Under the assumption of \textit{undercompleness}, a polynomial sample complexity is established for the RL algorithm for the infinite size of observation and action spaces, ensuring an $\epsilon-$suboptimal safe policy guarantee.

Application of Zone Method based Machine Learning and Physics-Informed Neural Networks in Reheating Furnaces

Despite the high economic relevance of Foundation Industries, certain components like Reheating furnaces within their manufacturing chain are energy-intensive. Notable energy consumption reduction could be obtained by reducing the overall heating time in furnaces. Computer-integrated Machine Learning (ML) and Artificial Intelligence (AI) powered control systems in furnaces could be enablers in achieving the Net-Zero goals in Foundation Industries for sustainable manufacturing. In this work, due to the infeasibility of achieving good quality data in scenarios like reheating furnaces, classical Hottel's zone method based computational model has been used to generate data for ML and Deep Learning (DL) based model training via regression. It should be noted that the zone method provides an elegant way to model the physical phenomenon of Radiative Heat Transfer (RHT), the dominating heat transfer mechanism in high-temperature processes inside heating furnaces. Using this data, an extensive comparison among a wide range of state-of-the-art, representative ML and DL methods has been made against their temperature prediction performances in varying furnace environments. Owing to their holistic balance among inference times and model performance, DL stands out among its counterparts. To further enhance the Out-Of-Distribution (OOD) generalization capability of the trained DL models, we propose a Physics-Informed Neural Network (PINN) by incorporating prior physical knowledge using a set of novel Energy-Balance regularizers. Our setup is a generic framework, is geometry-agnostic of the 3D structure of the underlying furnace, and as such could accommodate any standard ML regression model, to serve as a Digital Twin of the underlying physical processes, for transitioning Foundation Industries towards Industry 4.0.