Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations

This paper introduces tensorial calculus techniques in the framework of Proper Orthogonal Decomposition (POD) to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree $p$. Such nonlinear terms have an on-line complexity of $\mathcal{O}(k^{p+1})$, where $k$ is the dimension of POD basis, and therefore is independent of full space dimension. However it is efficient only for quadratic nonlinear terms since for higher nonlinearities standard POD proves to be less time consuming once the POD basis dimension $k$ is increased. Numerical experiments are carried out with a two dimensional shallow water equation (SWE) test problem to compare the performance of tensorial POD, standard POD, and POD/Discrete Empirical Interpolation Method (DEIM). Numerical results show that tensorial POD decreases by $76\times$ times the computational cost of the on-line stage of standard POD for configurations using more than $300,000$ model variables. The tensorial POD SWE model was only $2-8\times$ slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM shallow water equation model to compute its off-line stage faster than the standard and tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table

Autonomous robotic additive manufacturing through distributed model‐free deep reinforcement learning in computational design environments

AbstractThe objective of autonomous robotic additive manufacturing for construction in the architectural scale is currently being investigated in parts both within the research communities of computational design and robotic fabrication (CDRF) and deep reinforcement learning (DRL) in robotics. The presented study summarizes the relevant state of the art in both research areas and lays out how their respective accomplishments can be combined to achieve higher degrees of autonomy in robotic construction within the Architecture, Engineering and Construction (AEC) industry. A distributed control and communication infrastructure for agent training and task execution is presented, that leverages the potentials of combining tools, standards and algorithms of both fields. It is geared towards industrial CDRF applications. Using this framework, a robotic agent is trained to autonomously plan and build structures using two model-free DRL algorithms (TD3, SAC) in two case studies: robotic block stacking and sensor-adaptive 3D printing. The first case study serves to demonstrate the general applicability of computational design environments for DRL training and the comparative learning success of the utilized algorithms. Case study two highlights the benefit of our setup in terms of tool path planning, geometric state reconstruction, the incorporation of fabrication constraints and action evaluation as part of the training and execution process through parametric modeling routines. The study benefits from highly efficient geometry compression based on convolutional autoencoders (CAE) and signed distance fields (SDF), real-time physics simulation in CAD, industry-grade hardware control and distinct action complementation through geometric scripting. Most of the developed code is provided open source.</jats:p

A tensor decomposition approach to data compression and approximation of ND systems

The method of Proper Orthogonal Decompositions (POD) is a data-based method that is suitable for the reduction of large-scale distributed systems. In this paper we propose a generalization of the POD method so as to take the ND nature of a distributed model into account. This results in a novel procedure for model reduction of systems with multiple independent variables. Data in multiple independent variables is associated with the mathematical structure of a tensor. We show how orthonormal decompositions of this tensor can be used to derive suitable projection spaces. These projection spaces prove useful for determining reduced order models by performing Galerkin projections on equation residuals. We demonstrate how prior knowledge about the structure of the model reduction problem can be used to improve the quality of approximations. The tensor decomposition techniques are demonstrated on an application in data compression. The proposed model reduction procedure is illustrated on a heat diffusion proble