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Reliable data-driven models for engineering design

Reliable data-driven models for engineering design

Recently, with the universal development of AI and machine learning, data-driven methodologies have emerged and attracted much attention in engineering design. Provided sufficient data, these platforms put the burden of various parts of the design process (including finding governing equations, material models, and even solving the initial boundary value problems) on neural network functions of some form. These models are capable of learning real-life complexities and nonlinearities much better than traditional models which are usually bounded by numerous simplifications and linearizations. The enormous efficiency gains of data-driven methods are only beneficial if predictions are accurate and reliable for unseen scenarios. Conventional neural networks are notoriously weak in extrapolation (prediction outside the scope of training data), making them not trustworthy for engineering design.

One effective approach to obtain generalizable data-driven models is to construct neural network functions that, by definition, satisfy the desired physical laws. In this seminar, we introduce two of such neural network architectures: 1) Neural Peridynamic Operators (NPOs) for constitutive modeling of materials with complex behaviors; 2) Domain Agnostic Fourier Neural Operators (DAFNOs) as an efficient data-driven platform for engineering design.

For NPOs, we show that this architecture, which is based on peridynamics theory, satisfies balance laws. Then, we demonstrate the power and generalizability of this constitutive modeling tool for three different materials: single layer graphene, a hyperplastic anisotropic material, and a heterogenous biological tissue going under large deformation. For DAFNO, we redesign the architecture of Fourier Neural operators such that the efficient FFT-based integral evaluations remain valid and applicable on bounded arbitrary shaped domains, freeing this powerful method from the need to adhere to periodic rectangular domains. We show how the new model outperforms other state of the art neural operators via several examples in fluid and solid mechanics.

Bio: Dr. Siavash Jafarzadeh is a C.C. Hsiung Visiting Assistant Professor at Lehigh University’s department of mathematics. Prior to his appointment at Lehigh, he served as a Postdoctoral Scholar at Penn State University’s civil engineering department. Dr. Jafarzadeh received his PhD in mechanical engineering from University of Nebraska-Lincoln in 2021. His area of research includes computational mechanics, and mathematical and data-driven modeling with applications to damage mechanics and multiphysics. Dr. Jafarzadeh is one of the original developers of peridynamic models for corrosion damage and the fast convolution-based method for nonlocal models.

## Department of Mathematics and Statistics