Research

We are interested in combining machine learning with well-established numerical techniques for solving inverse problems in both solid and fluid mechanics. Presently, the focus is on the following problems:

• (Bayesian) Inference of discrepancy-augmented digital twin dynamical models

• Interpretable model discovery of nonlinear constitutive relations for hyperelastic and plastic materials

• Combining physics-based simulations with deep learning 

• Developing scalable and robust Bayesian inference for physics-augmented ML models

The concept of a digital twin involves creating a virtual framework that can replicate the behavior of a particular system over time. At the most basic level, a digital twin is a virtual duplicate of an engineering system built from a combination of models and measured data. Therefore, an important task in the development of a digital twin is the task of modelling the system mathematically. For a dynamical system, this modelling task entails deriving the governing equations of motion in the form of time-dependent differential equations. Typically, the mathematical models are unable to fully capture all possible dynamics exhibited by the physical system, due to the inherent simplifications like reduced-order modelling and approximations made when postulating the mathematical model. Hence, a discrepancy between the assumed model and the distribution of the observed data would always exist, and ignoring the discrepancy can lead to poor future predictions of the digital twin.

We are interested in learning the discrepancies from collected data (which could be measured data or simulated data from high-fidelity models) using machine learning techniques, such as (deep) neural networks, (deep) Gaussian processes, and generative models. 

In this work, the constitutive model of hyperelastic material is considered unknown apriori and the focus is on identifying admissible and interpretable constitutive models using machine learning techniques. We follow a Bayesian data-driven identification approach for discovering the constitutive law. To facilitate the model selection process, we employ a manually-designed library of polynomial basis functions inspired by extended Mooney-Rivlin models for rubber-like materials. A Bayesian sparse regression technique is employed to enforce a sparse selection of features from the library, utilizing a sparsity-promoting spike-and-slab prior. Rather than employing a computationally intensive Markov-Chain-Monte Carlo (MCMC) approach for Bayesian posterior inference, we adopt a more efficient and accurate Variational Bayesian (VB) approach, which significantly reduces the computational time. The approach will also be extended to plastic materials.

Recent works have shown that NN-based surrogate models achieve accuracies required for real-world, industrial applications while at the same time outperforming traditional solvers by orders of magnitude in terms of runtime. These success stories of machine learning approaches, deep learning (DL) in particular, have given rise to concerns that this technology has the potential to replace the traditional, simulation-driven approach to science. Here, we look to rely less on first principles and use data of sufficient size to provide the correct answers. Specifically, we are interested in the following question:

How can Deep Learning be used to enhance the analysis of Differential Equations in 

• • Solving high dimensional differential equations (e.g., multi-body physics)?

  • Solving highly parameterized ODEs and PDEs?

  • Solving inverse problems?