## Dynamical Reduced-Order Models for High-Dimensional Systems

Charous, A., 2023. *Dynamical Reduced-Order Models for High-Dimensional Systems*. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering and Center for Computational Science and Engineering, June 2023.

Advances in computational power have brought the possibility of realistically modeling our world with numerical simulations closer than ever. Nevertheless, our appetite for higher fidelity simulations and faster run times grows quickly; we will always grasp at what is just beyond our computational reach. No matter if we seek to understand biological, chemical, or physical systems, the bottleneck of scientific computing is almost always the same: high dimensionality. The goal of reduced-order modeling is to reduce the number of unknowns in a system while minimizing the loss of accuracy in the approximate solution. Ideal model-order reduction techniques are optimal compromises between computational tractability and solution fidelity. While there are plenty of such techniques to choose from, their widespread adoption remains to be seen due to several persistent challenges. Many methods are intrusive and difficult to implement, lack traditional numerical guarantees such as convergence and stability, and cannot adapt to unforeseen dynamics. We seek to promote the adoption of reduced-order models (ROMs) by creating non-intrusive, efficient, and dynamically adaptive algorithms that maintain the essential features and numerical guarantees of their full-order counterparts.

In this thesis, we derive and apply algorithms for dynamical reduced-order models. Many model-order reduction approaches project dynamical systems onto a fixed subspace obtained from either a simplification of the original equations, a set of known functions such as orthogonal polynomials, or a reduced basis of full-order simulations computed offline. However, if the true system exits the span of the prescribed subspace, such approaches quickly accumulate large errors. In contrast, dynamical ROMs adapt their subspaces as the system evolves. Geometrically, this amounts to integrating a dynamical system along a nonlinear manifold embedded in a full-order Euclidean space. We develop schemes that not only change subspaces at each discrete time step, but that change the subspace in between time steps for improved accuracy. Even further, our numerical schemes automatically detect when the dynamics depart the nonlinear manifold and may jump to a new nonlinear manifold that better captures the system state. For concreteness, we focus on a reduced-order modeling technique called the dynamical low-rank approximation (DLRA), a discrete analogue to the dynamically orthogonal (DO) differential equations. The DLRA evolves a low-rank system in time (or range) as an approximation to a full-rank system, and in contrast to many methods, the DLRA does not require an offline stage where full-order simulations are computed. It is also agnostic to the source of high dimensionality, whether it be the high resolution required, the large domain, or the stochasticity of the problem. These features make it a versatile tool suitable for a wide variety of problems. We evaluate, verify, and apply our new dynamical reduced-order models and schemes to a varied set of dynamical systems, including stochastic fluid flows and waves, videos and their dynamic compression, realistic ocean acoustics and underwater sound propagation with dynamic coordinate transforms, and stochastic reachability and time-optimal path planning.

The majority of this work is devoted to new adaptive integration schemes for the DLRA. We start by introducing perturbative retractions, which map arbitrary-rank matrices back to a manifold of fixed-rank matrices. They asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while guaranteeing convergence to the best low-rank approximation in a fixed number of iterations. From these retractions, we develop the dynamically orthogonal Runge-Kutta (DORK) schemes, which change the subspace onto which the system’s dynamics are projected in between time steps. The DORK schemes are improved by using stable, optimal (so) perturbative retractions, resulting in the so-DORK schemes. They are more efficient, accurate, and stable than their predecessors. We also introduce gradient-descent (gd) retractions and the gd-DORK schemes, which tend to converge rapidly to the best low-rank approximation by recursively applying retractions. The DORK schemes may be made rank-adaptive and robust to rank overapproximation with either a pseudoinverse or by changing the gauge of the integration scheme. While the pseudoinverse technique accumulates slightly more error, it preserves mode continuity, a feature that changing the gauge lacks. Next, we derive an alternating-implicit (ai) linear low rank solver, which is used to create ai-DORK schemes. The ai-DORK schemes are a general-purpose family of implicit integration schemes that have the same algorithmic complexity as explicit schemes (provided some conditions on the dynamics), which vastly broadens the scope of problems that can be solved with the DLRA. This relieves stringent time-step restrictions and enables the DLRA to handle stiff systems. Furthermore, we develop a piecewise polynomial approximation using adaptive clustering in order to handle non-polynomial nonlinearities in reduced-order models. We thoroughly

test these numerical schemes on well-conditioned and ill-conditioned matrix differential equations; data-driven dynamical systems including videos; Schrödinger’s equation; a stochastic, viscous Burgers’ equation; a deterministic, two-dimensional, viscous Burgers’ equation; an advection-diffusion partial differential equation (PDE); a nonlinear, stochastic Fisher-KPP PDE; nonlinear, stochastic ray tracing; and a nonlinear, stochastic Hamilton-Jacobi-Bellman PDE for time-optimal path planning. We find that the reduced-order solutions may be made arbitrarily accurate using rank adaptive dynamical schemes that automatically track the true rank of the full-order simulation, and nonlinearities may be well-approximated by dynamically increasing the number of stochastic clusters, all at a greatly reduced computational cost.

In addition to DORK schemes, we create a tailor-made low-rank integration scheme for the narrow-angle parabolic wave equation called the low-rank split-step Fourier method. Acoustic simulations are often bottlenecked by the Nyquist criterion, which insists that we sample spatially at least twice per wavelength. To address this, our low-rank split-step Fourier method has an algorithmic complexity that scales sublinearly in the number of classical degrees of freedom, enabling vastly larger computational domains and higher frequencies. We demonstrate its efficacy on realistic ocean acoustics problems in Massachusetts Bay with sound speed fields obtained from our high-resolution ocean primitive equations modeling system. In comparing the low rank and full-rank simulations, we demonstrate that the dynamical low-rank method captures the full-rank features including three-dimensional acoustic energy propagation in complex ocean fields with internal waves and rapidly varying bathymetry.

Lastly, with tools from machine learning, we introduce learnable and automatically differentiable coordinate transforms. The compressibility of a system heavily depends on the choice of coordinates, and frequently a coordinate system is chosen for its simplicity rather than its efficiency. Our novel coordinate transforms are determined in a hands-off manner by minimizing a cost function that includes the environmental data expressed in terms of the non-constant coefficients and initial conditions of a PDE. Not only do we automatically obtain Jacobians and Hessians of the transforms, we also find coordinate systems that reduce the rank of solutions to PDEs. This improves the accuracy of the DLRA for the same cost as a typical low-rank simulation, and it accelerates the convergence in rank to the full-order solution. The coordinate transforms also enable low-rank domain decomposition, which is particularly useful in ocean acoustics where the water-seabed interface is discontinuous. We demonstrate this methodology on a first-order PDE with advection and a second-order PDE, the parabolic wave equation, using two examples. We first show acoustic propagation along a three-dimensional wedge and compare the accuracy of solutions computed in the original and transformed coordinate systems. We then show acoustic propagation in a realistic ocean environment over Stellwagen Bank in Massachusetts Bay with a

dynamic coordinate transform.