Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We begin by introducing first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when there are multiple dynamics consistent with the data, these non-Euclidean adaptation laws implicitly regularize the learned model. Local geometry imposed during learning thus may be used to select parameter vectors—out of the many that will achieve perfect tracking or prediction—for desired properties such as sparsity. We apply this result to regularized dynamics predictor and observer design, and as concrete examples, we consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks. We subsequently develop a variational formalism based on the Bregman Lagrangian to define adaptation laws with momentum applicable to linearly parameterized systems and nonlinearly parameterized systems satisfying monotonicity or convexity requirements. We show that the Euler Lagrange equations for the Bregman Lagrangian lead to natural gradient and mirror descent-like adaptation laws with momentum, and we recover their first-order analogues in the infinite friction limit. We illustrate our analyses with simulations demonstrating our theoretical results.
We describe an efficient projection-based real-space implementation of the nonlocal single-determinant exchange operator. Through a matrix representation of the projected operator, we show that this scheme works equally well for both occupied and virtual states. Our scheme reaches a speedup of 2 orders of magnitude and has no significant loss of accuracy compared to an implementation of the full nonlocal single-determinant exchange operator. We find excellent agreement upon comparing Hartree-Fock eigenvalues, dipoles, and polarizabilities of selected molecules calculated using our method to values in the literature. To illustrate the efficiency of this scheme we perform calculations on systems with up to 240 carbon atoms.
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