Recently, machine learning (ML)-assisted models, such as neural networks, capable of describing some of the complex physical phenomena with good accuracy and reasonable computational cost are increasingly used in engineering applications. For exercises that involve many realizations of the engineering systems (e.g., uncertainty quantification, design under uncertainty), these ML-assisted models can be exploited here to develop physics-based surrogate models that are easy to evaluate once trained but at the same time accurate.However, these networks require a large dataset to train. In this research thrust, efficient training of neural networks using smaller datasets for applications to engineering problems are explored.

Our contributions are:

• Development of bi-fidelity Deep Operator Networks (BF-DeepONets) to model complex engineering systems (paper#1, paper#2).
  Applications:
  

• Development of transfer learning strategies for uncertainty quantification of complex engineering systems (paper). [codes]
  Application:
  

• Training of neural networks using l1-regularization and bi-fidelity data (paper).
  Application:
  

• Uncertainty quantification of locally nonlinear dynamical systems using neural networks (paper).
• Prediction of Ultrasonic Guided Wave Propagation in Solid-fluid and their Interface under Uncertainty using Machine Learning (paper).
• Application of the proposed strategies to multi-physics engineering problems.

The robust design of engineering systems requires the inclusion of uncertainties in the optimization process. The aim of this research thrust is to develop efficient design methodology and algorithms that can reduce the computational cost of robust and reliability-based optimization while considering uncertainty across multiple scales.

Topology Optimization under Uncertainty (TOuU)

In topology optimization (TO), we try to think about optimally distributing materials inside the structure to satisfy some performance criteria. However, in the presence of uncertainty, achieving a meaningful optimized design is computationally burdensome as the number of optimization variables is large in TO. In our recent works, we showed that the topology optimization under uncertainty for engineering design could be efficiently performed using multiple variants of the stochastic gradient descent algorithms (including two novel bi-fidelity algorithms), famously employed in the training of neural networks, but tailored for TO applications.
Our contributions are:

• Development of a stochastic gradient approach for TOuU (paper). [codes]
• Development of bi-fidelity stochastic gradient descent algorithms with proven linear convergence (paper).
• Applications: Topology optimization under micro-scale uncertainty, reliability-based topology optimization (paper#1,paper#2).

    

Optimal Design of Passive Structural Control Devices

In the recent past, many types of structures have been equipped with control devices to achieve some performance criteria (such as drift or acceleration mitigation). We developed computationally efficient design procedure of passive control devices for complex structures using NVIE approach.
The proposed method has the following characteristics (paper):

• Realizable computation time for large and complex structures.
• Trade-off between accuracy and speedup exists.
• Uncertainty in the existing structure can be incorporated.

Application:

Probabilistic Model Validation Framework

We developed a computationally efficient model validation framework applicable to models from vast domains based on philosophy advanced by the famous statistician George P. Box: ``Essentially, all models are wrong, but some are useful.'' This framework integrates the principle of falsification into the model selection process within a Bayesian framework utilizing measurement datasets from physical experiments to mitigate the weaknesses of existing individual validation schemes.
Our contributions are:

• Introduction of false discovery rate and likelihood-bound in model falsification (paper). [codes]
• A probabilistic machine learning framework is proposed for efficient validation of models (paper).

Applications:

Efficient Bayesian Model Selection

Bayesian model selection chooses, based on measured data, using Bayes’ theorem, suitable mathematical models from a set of possible models. In structural analysis, linear models are often used to facilitate design and analysis, though they do not always accurately reproduce actual structural responses. When the models also require the inclusion of nonlinearity to improve accuracy, the computation time required for response simulation increases significantly.
To address this issue, our contributions are (paper):

• Development of a computationally efficient method using Nonlinear Volterra type Integral Equations (NVIE) to model selection problems.
• Incorporating dynamic time history data for nonlinear models as the modal parameters changes with time in nonlinear models.
• Using NVIE approach the speedup is upto three orders of magnitude compared to traditional nonlinear solvers.
• The approach is demonstrated using a 100 DOF building structure subjected to earthquake excitation and a 1623 DOF three-dimensional building subjected to wind excitation.

Applications: