Boris Kramer

Research Summary

Our group is working on computational methods and numerical analysis for control, uncertainty quantification and optimization of complex and large-scale dynamical systems. Specifically, our research centers around reduced-order modeling and it's applications in many-query settings. Within reduced-order modeling we work on linear and nonlinear techniques and projection-based and fully data-driven, specifically on structure preservation (Lagrangian, Hamiltonian, gradient structure, port-Hamiltonian), system theoretic model reduction (e.g., balanced truncation, eigensystem realization) and also proper orthogonal decomposition. Within uncertainty quantification, we work on coherent risk estimation and certified risk-based design optimization (with conditional value-at-risk and buffered probabilities), reliability-based design optimization, multifideltiy UQ methods, and Bayesian inference. Working mainly in methods development, our research is interdisciplinary, both with respect to computational-science domains (computer science, engineering, mathematics) and applications (reactive and non-reactive fluids, cell biology, soft robotics, space weather, continuum mechanics, plasma physics, etc).

Research Topics

Reduced-order modeling

Reduced-order modeling describes a mathematical set of tools to reduce the complexity of high-dimensional systems via projection in the time domain, interpolation in the frequency domain, or by learning reduced-order models purely from data. Specifically, our research centers around:

System-theoretic model reduction for control: This branch of model reduction uses classical principles such as controllability/observability, pole/residue approximations, positive-realness and others to derive reduced-order models, mostly for controlled (open or closed-loop) systems. Our work mostly focused on nonlinear balanced truncation model reduction ([J19, J35, C12, PP01, PP03]) as well as eigensystem realization ([J01, J07]) to learn ROMs from impulse response data. Using system-theoretic model reduction is often the key to enabling reduced-order controllers, as the methods are not prone to a dependence on simulated data, but only use the model's governing equations to derive the low-dimensional models.

Structure-preserving Operator Inference: Reduced-order modeling provides us with a mathematically sound framework to integrate data and physics in computationally efficient surrogate model learning. Operator Inference (see the 2024 survey [J32]) uses simulated, observed, or experimental data to infer low-dimensional subspaces or manifolds where high-dimensional solutions can be well approximated. The structure of the ROM operators leverages physical insights, such as the polynomial degree of mechanisms, conservation principles, etc. Our work explores the possibilities of embedding Lagrangian structure of second-order systems (see publications [J33, J34, C11, C13]), Hamiltonian principles ([J21, J29, J36]), stability guarantees ([J18, J25]), shift-invariance ([J23]), and others (to come).

Applications: The goal of our work is to advance progress on reduced modeling of challenging applications, such as rocket combustion (J12, C08, C09]), soft robotics ([C13, C11, J32]), space weather ([J23, J30]), and others. This allows us to contribute in two exciting ways: A) advance ROMs for realistic applications where they are most needed; B) identify and work on new research questions arising with increased model complexity, which naturally feeds back into fundamental research questions.

Multifidelity uncertainty quantification (UQ) and design under uncertainty

In the presence of parametric and model form uncertainty, quantifying and managing uncertainties is key for decision makers and engineers. One aspect of this area focuses on computing statistical information such as mean and variance, or other information about the distribution of quantities of interest, such as coherent risk measures (CVaR, buffered probabilities, see [J20]). Traditionally, this would often require a large number of expensive model evaluations, rendering high-fidelity models infeasible to be used for this task. Multifidelity UQ leverages information from models of varying fidelity and computational cost to efficiently solve the UQ task at hand. Such methods are attractive as they move most of the computational work to lower fidelity models and decrease the number of expensive high-fidelity model evaluations. In this area, our research connects often with reduced-order modeling and more generally statistical surrogate modeling, as follows:

Coherent risk measures for certifable design optimization: Coherent risk measures follow a set of axioms to arrive at a principled way to estimate tail risk ("black swans", "unwanted outcomes") which can often derail projects and cause catastrophic failure. By their nature of being rare, tail measures are hard to evaluate. We have developed a suite of surrogate-model-assisted risk estimators based on reduced-order models ([J09, J15]) and polynomial dimensional decomposition (PDD) for dependent random inputs ([J26, J27]). Moreover, we worked on design with risk measures ([J20, C07])

Bayesian Inference: The Bayesian perspective to inference starts with an intitial belief state and updates this belief as new information becomes available. These updates require a model evaluation --- which again can be very expensive. We have worked on Bayesian Inference and multimodel inference in the context of systems biology ([J24, PP02]), composite failure modeling ([J28]), solar wind modeling ([J30]) and Bayesian model learning of Hamiltonian systems ([J40, C10]).

We have embedded this quantified parametric uncertainty in reliability-based design optimization (RBDO) and design under uncertainty. There, multifidelity approaches save orders of magnitudes of computational cost when evaluating probabilistic constraints within the design optimization loop. Moreover, work on reusing information from past design iterations has also helped reduce the computational cost of RBDO.

Variable transformations and lifting

Evolutionary processes in engineering and science are often modeled with nonlinear non-autonomous ordinary differential equations that describe the time evolution of the states of the system, i.e., the physically necessary and relevant variables (c.f., the short survey [J22]). However, these models are not unique: the same evolutionary process can be modeled with different variables, which can have a tremendous impact on computational modeling and analysis. This idea of variable transformation (referred to as lifting when extra variables are added) to promote model structure is found across different communities, with literature spanning half a century. Our group is interested in exploring variable transformations for ROM learning ([J12, J14, C06, C08, C09]) discovery and exploitation of structure ([J19]) and especially designing algorithms and provable guarantees for finding such transformations ([J31]).

Current Projects

Prediction, damage analysis and risk assessment for a gas power plant via fast and accurate reduced models. (Korea Institute for Advancement of Technology (KIAT) through the International Cooperative R&D program (No. P0019804, Digital twin based intelligent unmanned facility inspection solutions).; 12/01/2021-11/30/2024). This project involved BS & MS student Elle Lavichant, Postdocs Dongjin Lee and Harsh Sharma, as well as graduate student Hyeonghun Kim.

Collaborative Research: Nonlinear Balancing: Reduced Models and Control (NSF Grant 2130727; 01/01/2022-12/31/2024; in collaboration with Profs. Jeff Borggaard and Serkan Gugercin, Virginia Tech). In this project, we develop a new class of reduced-order models and controllers for complex high-dimensional polynomial nonlinear systems via the concept of nonlinear balanced truncation. Specifically, we developed a scalable tensor-based approach to solve the HJB equations required for nonlinear balancing for a class of polynomial control-affine systems, to obtain polynomial expansions of the energy functions required for balanced truncation, as well as high-performance algorithms and numerical analysis to analyze the conditioning of the tensorized problems. Additionally, reduced-order nonlinear controllers are designed using a simultaneous reduction and control framework, which is far superior to the existing reduce-then-control framework. Phd Student Nick Corbin is working on this project.

CAREER: Goal-oriented Variable Transformations for Efficient Reduced-order and Data-driven Modeling (NSF Grant 2144023; 05/01/2022-04/30/2026). In this project, we develop the foundations of a new theoretical and computational paradigm that leverages variable transformations to uncover low-dimensional structures in nonlinear dynamical systems and achieve efficient and accurate model reduction that may be certified with respect to stability and structure-preservation. We approach this in model- and data-driven settings by designing symbolic computing algorithms that systematically identify transformations and subsequent order-reduction projections that result in optimal quadratic or polynomial models that also preserve symplectic structure for Hamiltonian systems. In the data-driven case, transformations are sought that lead to long-term predictive reduced-order models that are physically interpretable and have favorable numerical properties. Through this effort, new low-dimensional models of the physics of medium-scale applications of chemical reaction dynamics and additive manufacturing will be discovered. The methodological contributions will be assessed on large-scale models of reactive flows and ocean dynamics. PhD student Albani Oliveiri and undergraduate student Anique Dittrich are working on this project.

Nonlinear Data-driven and Structure-Preserving Hamiltonian Model Reduction (ONR Grant N00014-22-1-2624; 08/01/2022-07/31/2025). Computational modeling, simulation and control of physical systems characterized by Hamiltonian mechanics abound in naval applications, such as ocean flow modeling, plasma physics, and continuum mechanics models that follow a principle of least action. From a mathematical perspective, Hamiltonian systems have additional physical and geometric structures in the form of symmetries, symplecticity, first integrals, and energy preservation. Those properties need to be preserved in time and space discretization, and particularly in data-driven reduced-order modeling, which this project is concerned with. Specifically, we are developing nonlinear structure-preserving data-driven reduced-order modeling strategy for Hamiltonian systems. At UCSD, Postdoc Harsh Sharma is working on this project, with contributions from undergraduate student Juan-Diego Draxl. Our collaborator Zhu Wang (U. South Carolina) is a subaward on this project. Journal publications [29, 32, 33, 34, 36, 40] and conference publications [10, 11, 13] reflect this project.

Reduced-Order Modeling for Real-time Simulation of Flow Phenomena in Semiconductor Manufacturing. (Samsung Electronics Co.).; 03/15/2024-03/30/2025). This proposed collaborative project provides both training to the Visting Industry Fellow Dr. Seunghyon Kang as well as new research in surrogate modeling for flow phenomena occurring during semiconductor manufacturing. We will mainly focus on data-driven reduced-order modeling via operator inference. Postdoc Harsh Sharma and graduate student Hyeonghun Kim are partly contributing to this project.

MURI: Mathematics of Digital Twins. (Air Force Office of Scientific Research.; 09/2024-10/2029). For a high-level overview, see the UC San Diego Today article . In this project, we will develop task-oriented (parametric and system-theoretic) reduced-order models for digital twin development, with a focus on metal-additive manufacturing.

ACCORD: AFRL–UCSD Collaborative Center for Optimal Risk-quantified and robust Design of aerospace vehicles. (Air Force Research Lab.; 12/08/2023-09/07/2029). The ACCORD center will revolutionize the design of next-generation aerospace vehicles through new computational design methods centering around large-scale multidisciplinary design optimization (MDO); for a high-level overview, see the UCSD News article New Center Supports Next-Gen Air Force Vehicle Design. The center is led at UCSD by Professor John Hwang, and co-PIs are Professors Oliver Schmidt and J.S. Chen (SE). Professor Markus Rumpfkeil at U. of Dayton is also collaborating on this project. Our research group will contribute with reduced-order modeling for coupled systems as well as by providing new tools for surrogate-assisted risk-based design optimization at scale.

Past Projects

SWQU: Composable Next Generation Software Framework for Space Weather Data Assimilation and Uncertainty Quantification (NSF Award 2028125; 09/01/2020-08/31/2024). In this collaborative project with experts in geospace sciences (R. Linares, MIT; A. Ridley, UMich; Phil Erickson, MIT Haystack), uncertainty quantification (Y. Marzouk, MIT), fluid mechanics (J. Peraire, MIT), and software development (A. Edelman, MIT) we developed a variety of computational models and techniques for space weather prediction; see the UCSD news article Making space weather forecasts faster and better. Another article about space weather modeling that mentions our work appeared in Science News Solar storms can wreak havoc. We need better space weather forecasts. Journal publications [30,31, 39] were produced as part of this project. Phd student Opal Issan spent her first three years on this project, working on reduced-order modeling and uncertainty quantification for solar wind propagation.

Multifidelity Risk Assessment of High-Performance Systems. (08/20/2021-12/31/2022). In this project we developed surrogate-assisted methods for estimating coherent risk measures, such as conditional value-at-risk for complex engineering systems. We proposed bifidelity and multifidelity techniques that work in the context of depenent random variables. The surrogate models were developed as the dimensionally-decomposed generalized polynomial chaos expansions, with novel twists on how to train them effectively and use them in risk estimation. Journal publications [26, 27] resulted from this project. Postdoc Dongjin Lee worked on this project.

Newton Award for Transformative Ideas during the COVID-19 Pandemic (07/01/2020-12/31/2020). This 6-month project together with Prof. Melvin Leok (UCSD, Mathematics) developed ideas for a novel framework and mathematical formulation for systematic, robust, and efficient multi-fidelity modeling of large-scale hierarchical interconnected systems. We explored the combination of geometric structure-preserving numerical integration and model reduction in the context of hierarchical interconnected systems. Refer to the news article Mathematician, Engineer receive Newton Award for Transformative Ideas during COVID-19 Pandemic for more info.

SBIR Phase I: Human-Centered, Augmented Intelligence Software for Water and Wastewater (NSF Award 2004275 via Confluency LLC, Chicago; 07/01/2020-06/31/2020). This Small Business Innovation Research (SBIR) Phase I project developed methods for combining multi-fidelity simulation models and data-driven models to support decision-making for both long-term planning needs and real-time operational decision support for water and wastewater systems. Specifically, we investigated the reduced-order modeling for the St Venant equations for the one- and two-dimensional flow in waterways. The figure on the left shows a Lagrangian particle simulation for a storm wave travelling through a channel. MS student Liezl Maree worked on this project.