Session: 13-22-01: CONCAM Distinguished Lectures on Computational Mechanics I

Paper Number: 173642

Computational Mechanistic Intelligence Based on Hierarchical Deep-Learning Neural Networks-Artificial Intelligence (Hidenn-Ai): Part Ii 

In this two-part talk, I will present Hierarchical Deep Learning Neural Networks for Artificial Intelligence (HIDENN-AI), an integrated, data-free framework for solving governing partial and ordinary differential equations (PDEs/ODEs) that arise in a broad range of science and engineering applications. HIDENN-AI establishes a hierarchical and structured approach that directly incorporates deep neural network (DNN) architectures into the weak (Galerkin) form of the governing differential equations. This framework effectively bridges the gap between traditional numerical methods and modern machine learning, providing both flexibility and computational scalability. The foundation of HIDENN-AI lies in its use of three elementary computational building blocks—linear transformation, multiplication, and inversion—that collectively enable DNN-based representations of widely used interpolation functions. These include those derived from the finite element method (FEM), Lagrangian polynomial basis functions, spline and B-spline approximations, reproducing kernel meshfree shape functions, non-uniform rational B-splines (NURBS), and basis functions used in isogeometric analysis (IGA). This representation enables HIDENN-AI to be compatible with various established discretization strategies while leveraging the flexibility and adaptability of neural networks.

A distinguishing feature of HIDENN is that the neural network parameters, i.e., the weights and biases are formulated as functions of nodal positions. As a result, the training process in HIDENN does not only solve for the unknown field variables that satisfy the governing differential equations, but it also automatically optimizes the distribution of nodal points and mesh configuration, a capability known as r-adaptivity. This coupling of solution and spatial discretization within a single training process represents a key aspect of the Software 2.0 paradigm, differentiating HIDENN-AI from traditional solvers or black-box machine learning approaches. It eliminates the need for pre-generated training datasets and allows the network to generalize well across problem domains.

Building on the HIDENN foundation, we introduce an extended formulation called Convolutional HIDENN (C-HiDeNN). This extension incorporates convolutional operations into the hierarchical network architecture, significantly improving the representation of shape function derivatives, enabling structured adaptivity, and handling complex material derivatives more accurately. The C-HiDeNN framework also expands the parameter space for optimization by introducing variables such as polynomial order (p), dilation factor (a), patch size (s), and nodal positions (X). These additional degrees of freedom allow the model to implement a fully meshfree and adaptive discretization, which can be tailored to specific physics or application needs in real-time.

To address the computational challenges of high-dimensional problems, we further introduce C-HiDeNN with Tensor Decomposition (C-HiDeNN-TD). This formulation employs tensor decomposition techniques to separate an n-dimensional PDE into n one-dimensional sub-problems, significantly reducing the computational burden in terms of both processing time and memory requirements. This approach enables C-HiDeNN-TD to scale to ultra-large problem sizes, making it particularly well-suited for high-resolution simulations in complex geometries and parameterized domains. Unlike most machine learning-based solvers, C-HiDeNN-TD does not require any labeled training data. Instead, it serves as a physics-informed surrogate model that can accurately approximate solutions across a space-parameter-time (SPT) domain with minimal computational overhead.

We demonstrate the effectiveness of HIDENN-AI through several numerical examples. These include applications in solid mechanics, where stress-strain relationships under complex loading conditions are modeled accurately; additive manufacturing, where thermal and mechanical fields are simulated during layer-by-layer deposition; and topology optimization, where the spatial distribution of material is optimized concurrently with the field solution. These case studies highlight the framework’s ability to provide high-fidelity solutions with efficient resource usage, as well as its potential for integration into design and control systems.

In conclusion, HIDENN-AI offers a versatile, interpretable, and computationally efficient approach for solving complex, nonlinear, and high-dimensional differential equations. By integrating deep learning with traditional numerical analysis in a principled and physics-consistent manner, HIDENN-AI opens new directions for data-free simulation, design optimization, and real-time control in a variety of engineering domains.

Presenting Author: Dong Qian University of Texas At Dallas

Presenting Author Biography: Dr. Dong Qian is professor and associate department head of mechanical engineering at the University of Texas at Dallas. He received his B.S. degree in Bridge Engineering from Tongji University in China in 1994, his M.S. degree in Civil Engineering from the University of Missouri in 1998 and Ph.D. degree in Mechanical Engineering from Northwestern University in 2002. He started his academic career as an Assistant Professor of mechanical engineering at the University of Cincinnati and was promoted to Associate Professor with tenure in 2008. In the Fall of 2012, he joined the newly established Mechanical Engineering Department as a tenured Associate Professor at the University of Texas at Dallas and was promoted to full Professor in 2015. Dr. Qian has conducted research and published extensively in the general areas of computational mechanics and materials, including nonlinear meshfree and particle methods, hierarchical and concurrent multiscale methods, computational nanomechanics, fatigue and simulation-based life prediction, surface engineering, additive manufacturing, data-driven modeling and simulation and applications of machine learning. His work has been widely cited (with > 13000 citations) and appeared in high-impact journals such as Science, Nature Communication, Advanced Materials, Applied Mechanics Reviews, Journal of Applied Mechanics, Computer Methods in Applied Mechanics and Engineering. Dr. Qian is a fellow of the American Society of Mechanical Engineering. He currently serves as a member of the editorial board for the Journal of Computational Mechanics and is an associate editor for the Journal of Computer Modeling in Engineering and Sciences.

Authors:

Dong Qian University of Texas At Dallas
Yingjian Liu The University of Texas at Dallas
Jiachen Guo Northwestern University
Chanwook Park Northwestern University
Gino Domel Northwestern University
Hantao Zhang Northwestern University
Monish Pabbala The University of Texas at Dallas
Wing Kam Liu Northwestern University