where X is Fluid Dynamics

| Aerospace | Climate | Biomed | Energy |

A digital twin is defined as a virtual representation of a physical asset or a process enabled through data and simulators for real-time prediction, optimization, monitoring, control, and informed decision-making. Our lab is working to tap into the potential of this technology by developing advanced computational methods for complex nonlinear dynamical systems driven by multiscale and multiphysics processes. Our current focus is on exploring the essential components necessary to achieve fast, scalable, and generalizable methods for scientific machine learning (SciML).

The systems studied in our lab range from engineering flows to geophysical flows with different levels of fidelities. Our current SciML efforts are centered in the development of hybrid analysis and modeling (HAM) approaches in LES and ROM frameworks, which is particularly relevant given the emphasis on modeling and analysis of turbulent flows across a variety of spatial and temporal scales. Integrating different ways of physics-based and machine learning/data-driven modeling approaches, we advocate hybrid modeling approaches for emerging digital twin technologies, and exploit our HAM framework that embraces flexibility to use most effective tools depending on the problem and its challenges.

### Hybrid Analysis and Modeling (HAM)

Quo vadis in modeling: physics-based or data-driven? One camp develops and improves first principle models that provide the best generalizability and trustworthiness characteristics. In the other camp, researchers try to explain phenomena from archival data using statistical approaches. Compared to the physics-based modeling approach, these models thrive on the assumption that data is a manifestation of both known and unknown physics and hence when trained with an ample amount of data, the data-driven models might learn the full physics on their own. This approach, involving in particular deep learning, has started achieving human-level performance in several tasks. Some of the advantages of these data-driven models are online learning capability, computational efficiency for inference, accuracy even for very challenging problems as far as the training, validation and test data are prepared properly. However, due to their data-hungry and black-box nature, poor generalizability, inherent bias, and lack of robust theory for the analysis of model stability, their acceptability in multi-scale and multi-physics systems is fairly limited. To address such challenges, we explore a new HAM approach based on a synergistic combination of deterministic and statistical model components.

### Large Eddy Simulations (LES)

The goal of large eddy simulation, which is one of the most successful approaches for simulating turbulent flows, is to decompose the flow into large and small scales by convolving the flow with a spatial low-pass filter. This process results in a well-known closure problem due to the nonlinearity of the underlying governing equations. Currently, we are working on functional, structural and data-driven closure modeling ideas to take into account subgrid-scale effects in large eddy simulations. Closure modeling refers to the process of including the truncated scales (due to the limited numerical resolution & computational resources) into the resolved dynamics (the scales captured in our underlying numerical model) to account for the missing subgrid-scale physics, which is quite important if the underlying dynamical process is nonlinear with strong interactions between small and large scales.

### Reduced Order Modeling (ROM)

Reduced order models are extremely low-dimensional models that can decrease the computational cost of current computational models by orders of magnitude. Simplifying computational complexity of the underlying mathematical model, these models offer promises in settings, especially where the traditional methods require repeated model evaluations over a large range of parameter values. To develop such low dimensional models that are accurate in realistic problems, the closure problem needs to be solved, i.e., the effect of the discarded modes on the model dynamics needs to be modeled.

## Sponsors

We are grateful for the support from DOE, NSF, NASA, ASHRAE, NVIDIA, and RCN.