Stefanie Günther

Stefanie Günther, N. Anders Petersson

We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ real-arithmetic solvers by representing complex matrices in term of their real and imaginary parts. While the reconstruction is trivial when the spectrum of the real-valued embedding is simple, degenerate and/or complex conjugated eigenvalues introduce ambiguities because each eigenspace may include contributions from both the unitary matrix and its complex conjugate. We prove that this ambiguity can always be resolved by applying a structured projection to the eigenspaces of the real-valued embedding, followed by a rank-revealing orthonormalization. The resulting procedure recovers the eigenvalues and an unitary eigenbasis for the original unitary matrix, with correct multiplicities of degenerate eigenvalues.

Peter Sentz, Stanley Nicholson, Yujin Cho et al.

The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.

Stefanie Günther, Will Pazner, Dongping Qi

Based on the continuous interpretation of deep learning cast as an optimal control problem, this paper investigates the benefits of employing B-spline basis functions to parameterize neural network controls across the layers. Rather than equipping each layer of a discretized ODE-network with a set of trainable weights, we choose a fixed number of B-spline basis functions whose coefficients are the trainable parameters of the neural network. Decoupling the trainable parameters from the layers of the neural network enables us to investigate and adapt the accuracy of the network propagation separated from the optimization learning problem. We numerically show that the spline-based neural network increases robustness of the learning problem towards hyperparameters due to increased stability and accuracy of the network propagation. Further, training on B-spline coefficients rather than layer weights directly enables a reduction in the number of trainable parameters.

Eric C. Cyr, Stefanie Günther, Jacob B. Schroder

This paper investigates multilevel initialization strategies for training very deep neural networks with a layer-parallel multigrid solver. The scheme is based on the continuous interpretation of the training problem as a problem of optimal control, in which neural networks are represented as discretizations of time-dependent ordinary differential equations. A key goal is to develop a method able to intelligently initialize the network parameters for the very deep networks enabled by scalable layer-parallel training. To do this, we apply a refinement strategy across the time domain, that is equivalent to refining in the layer dimension. The resulting refinements create deep networks, with good initializations for the network parameters coming from the coarser trained networks. We investigate the effectiveness of such multilevel "nested iteration" strategies for network training, showing supporting numerical evidence of reduced run time for equivalent accuracy. In addition, we study whether the initialization strategies provide a regularizing effect on the overall training process and reduce sensitivity to hyperparameters and randomness in initial network parameters.