Artificial Neural Network Approach to the Analytic Continuation Problem

Inverse problems are encountered in many domains of physics, with analytic continuation of the imaginary Green's function into the real frequency domain being a particularly important example. However, the analytic continuation problem is ill defined and currently no analytic transformation for solving it is known. We present a general framework for building an artificial neural network (ANN) that solves this task with a supervised learning approach. Application of the ANN approach to quantum Monte Carlo calculations and simulated Green's function data demonstrates its high accuracy. By comparing with the commonly used maximum entropy approach, we show that our method can reach the same level of accuracy for low-noise input data, while performing significantly better when the noise strength increases. The computational cost of the proposed neural network approach is reduced by almost three orders of magnitude compared to the maximum entropy method.

Calibration of high-precision flexure parallel robots

Nuno Ricardo Fazenda Carrico

Over the last decades, calibration techniques have been widely used in robotics since they represent a cost-effective solution for improving the accuracy of robots and machine-tools. They only involve software modification without the necessity of revising the robot design or tightening the manufacturing tolerances. The goal of this thesis is to propose a procedure that guides the engineer through the calibration of a given multi-DOF flexure parallel robot within sub-µm accuracy. Two robots having 3 and 6 degrees of freedom have been considered as a case-study throughout the work. As in any calibration procedure, the work has been conducted on three different fronts: measurement, data processing and validation. The originality of this thesis in respect to published material lies in these three points. Measurements were carried out in a chamber inside which the measuring environment was protected against mechanical and thermal perturbations. In particular, the temperature variations experienced by the different parts of the measuring loop during a typical measurement session were stabilized within less than ± 0.1 °C. Proposed procedures allow the collection of reliable sets of data on the two robots. Delicate aspects of practical implementation are discussed. In particular, the problem of collecting a complete set of 6D data within accuracies in the nanometre range, for which there is still a lack of standard equipment, is solved using a procedure comprising several steps and making use of existing instrumentation. Suggestions for future investigations are given, regarding either long-term research problems or short-term industrial implementation issues. Data processing was performed using two different techniques in order to reach absolute accuracies after calibration better than ± 100 nm for translations and ± 3 arcsec for rotations (± 0.3 arcsec inside a more restricted range of ± 0.11°). The first method is called the "model-based approach" and requires the use of a known analytical relationship between the motor and operational coordinates of the robot. This relationship involves a certain number of parameters that can be related to the geometry of the robot (physical models) or simply mathematical coefficients of an approximating mathematical function (behavioural models). In the case of high-precision multi-DOF flexure parallel robots, we show that polynomial-based behavioural models are preferable to physical models in terms of accuracy for data processing tasks. In the second method, called the "model-free approach", the user does not need to model explicitly the main error sources (or their effect) affecting the robot accuracy. A model-free approach has been implemented using Artificial Neural Networks. We show that, using a heuristic search based on a decision-tree, the architecture of a network with satisfactory prediction capability can be found systematically. In particular, this algorithm can find a network able to predict the direct correspondence between the motor and operational coordinates (within the desired accuracy) without the help of the Inverse Geometric Model of the robot, i.e. even if the nominal geometry of the robot being calibrated remains unknown. This result contradicts conclusions reported by previous researchers. It is claimed that any robot (not necessarily a high-precision flexure parallel mechanism) can be calibrated by means of a "neural approach" in which the architecture of an appropriate network is determined with the help of our algorithm. Two examples (other than the robots measured in this thesis) are given to illustrate this universality. In the last part of this work, we provide a feasibility study on the use of indentation, a technique traditionally used for material testing, as a validation procedure to assess the accuracy of the calibrated degrees of freedom. The industrial interest of this technique lies in the fact that the robot is asked to execute similar motions to those involved in a real micro-machining operation.

EPFL2007

Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems

Alexandra Nagy, Vincenzo Savona

The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom (d.o.f.) is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the nonequilibrium steady state of Markovian open quantum systems based on variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative XYZ spin model on a lattice.

AMER PHYSICAL SOC2019

Inverse Modelling and Predictive Inference in Continuum Mechanics: a Data-Driven Approach

Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.