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Optimization with learning-informed differential equation constraints and its applications
Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
First-order conditions for the optimal control of learning-informed nonsmooth PDEs
In this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the corresponding optimal control problem. Finally, several stationarity conditions are derived based on different notions of generalized differentiability.
Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems
Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.
Optimization with learning-informed differential equation constraints and its applications
Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented.
A physically oriented method for quantitative magnetic resonance imaging
Quantitative magnetic resonance imaging (qMRI) denotes the task of estimating the values of magnetic and tissue parameters, e.g., relaxation times T1, T2, proton density p and others. Recently in [Ma et al., Nature, 2013], an approach named Magnetic Resonance Fingerprinting (MRF) was introduced, being capable of simultaneously recovering these parameters by using a two step procedure: (i) a series of magnetization maps are created and then (ii) these are matched to parameters with the help of a pre-computed dictionary (Bloch manifold). In this paper, we initially put MRF and its variants in the perspective of optimization and inverse problems, providing some mathematical insights into these methods. Motivated by the fact that the Bloch manifold is non-convex, and the accuracy of the MRF type algorithms is limited by the discretization size of the dictionary, we propose here a novel physically oriented method for qMRI. In contrast to the conventional two step models, our model is dictionary-free and it is described by a single nonlinear equation, governed by an operator for which we prove differentiability and other properties. This non-linear equation is efficiently solved via robust Newton type methods. The effectiveness of our method for noisy and undersampled data is shown both analytically and via numerical examples where also improvement over MRF and its variants is observed.