Research Areas:

Manifold Learning in Nonlinear Mechanics and Materials

An Interpretable Machine Learning Model for Deformation of Multi–Walled Carbon Nanotubes

We present a novel interpretable machine learning model to predict accurately the complex rippling deformations of Multi–Walled Carbon Nanotubes (MWCNTs) made of millions of atoms. The proposed model accurately matches an atomistic–physics–based model while being orders of magnitude faster. It extracts universally dominant patterns of deformation in an unsupervised manner. These patterns are comprehensible and explain how the model predicts, yielding interpretability.

Upendra Yadav, Shashank Pathrudkar, Susanta Ghosh, "Interpretable machine learning model for the deformation of multiwalled carbon nanotubes", Physical Review B 103 (3), 035407, 2021. Read more

Physics Informed Neural Network

We propose a novel PINN scheme that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, we have used the Cahn Hilliard and Allen Cahn equations, which are widely used to describe phase separation and reaction diffusion systems. Our results show significant improvement in accuracy over the PINN method while using a smaller number of collocation points.

R. Mattey, S. Ghosh"A Physics Informed Neural Network for Time–Dependent Nonlinear and Higher Order Partial Differential Equations", Read more

Machine learning based prediction of the electronic structure of quasi-one-dimensional materials under strain

We present a machine learning based model that can predict the electronic structure of quasi-one-dimensional materials while they are subjected to deformation modes such as torsion and extension/compression. Using armchair single wall carbon nanotubes as a prototypical example, we demonstrate the use of the model to predict the fields associated with the ground state electron density and the nuclear pseudocharges, when three parameters - namely, the radius of the nanotube, its axial stretch, and the twist per unit length - are specified as inputs. We comment on the interpretability of our machine learning model and discuss its possible future applications.

Pathrudkar, S., Yu, H. M., Ghosh, S., & Banerjee, A. S. (2022). Machine learning based prediction of the electronic structure of quasi-one-dimensional materials under strain. Physical Review B, 105, 195141. Read more

Atomistic-Continuum Modeling for Low Dimensional Materials

An Atomistic-based Finite Deformation Continuum Membrane Model for Monolayer Transition Metal Dichalcogenides

A finite-deformation crystal-elasticity membrane model for Transition Metal Dichalcogenide (TMD) monolayers is presented. Monolayer TMDs are multi-atom-thick two-dimensional (2D) crystalline membranes having atoms arranged in three parallel surfaces. In the present formulation, the deformed configuration of a TMD-membrane is represented through the deformation map of its middle surface and two stretches normal to the middle surface. Crystalelasticity based kinematic rules are employed to express the deformed bond lengths and bond angles of TMDs in terms of the continuum strains. The continuum hyper-elastic strain energy of the TMD membrane is formulated from its inter-atomic potential. The relative shifts between two simple lattices of TMDs are also considered in the constitutive relation. A smooth finite element framework using B-splines is developed to numerically implement the present continuum membrane model. The proposed model generalizes the crystal-elasticity based membrane theory of purely 2D membranes, such as graphene, to the multi-atom-thick TMD crystalline membranes. The significance of relative shifts and two normal stretches are demonstrated through numerical results. The proposed atomistic-based continuum model accurately matches the material moduli, complex post-buckling deformations, and the equilibrium energies predicted by the purely atomistic simulations. It also accurately reproduces the experimental results for large-area TMD samples containing tens of millions of atoms.

U. Yadav and S. Ghosh, "An atomistic-based finite deformation continuum membrane model for monolayer Transition Metal Dichalcogenides," Journal of the Mechanics and Physics of Solids, 168 (2022) 105033, August 2022. 

An Atomistic-based 3D Foliation Model for Multilayer Graphene Materials and Nanotubes

We present a three-dimensional continuum model for layered crystalline materials made out of weakly interacting two-dimensional crystalline sheets. The constitutive model for the bulk is derived from the atomistic interactions by appropriate kinematic assumptions, adapted to the foliation structure and mechanics. We find that the new model is very efficient and accurate. 

S. Ghosh and M. Arroyo, “An Atomistic-based 3D Foliation Model for Multilayer Graphene Materials and Nanotubes," Journal of the Mechanics and Physics of Solids. 61, 2013, pp. 235-253. Read More

Inverse Problems

Modified error in constitutive equations (MECE) approach for ultrasound elastography

A partial differential equation-constrained optimization approach is presented for reconstructing mechanical properties (e.g., elastic moduli). The proposed method is based on the minimization of an error in constitutive equations functional augmented with a least squares data misfit term referred to as MECE for “modified error in constitutive equations.” The main theme of this paper is to demonstrate several key strengths of the proposed method on experimental data. In addition, some illustrative examples are provided where the proposed method is compared with a common shear wave elastography (SWE) approach. 

S. Ghosh, Olalekan Babaniyi, M. Diaz, Z. Zou, M. Bayat, M. Fatemi and Wilkins Aquino, “Modified Error in Constitutive Equations (MECE) Approach for Ultrasound Elastography,” The Journal of the Acoustical Society of America, Vol. 142, No. 4, Oct 2017. DOI:10.1121/1.5006911 Read More

Uncertainty Quantification of Bayesian Neural Networks

A Bayesian neural network with probabilistic weights and biases.

Images with low uncertainty show separation of two classes in a low dimensional projection whereas, the images with high uncertainty are randomly spread without any separation. 

Machine learning models have enormous potential in the field of medicine ranging from fundamental applications such as image classification, segmentation, and information extraction to more complex applications such as therapeutic discoveries and personalized treatments.   Though machine-assisted decision-making in such applications can have radical impacts, errors can be catastrophic. Thus, quantifying uncertainties in machine decisions should be given utmost importance in medical applications.  

We develop and analyze Bayesian neural network models primarily for medical image classification with the aim to quantify uncertainties in their prediction. Further, we utilize the uncertainties obtained by the Bayesian neural network to improve the network's performance on a large subset of data. We provide an explanation for these uncertainties by performing a non-linear dimensional reduction on the feature space. We have shown that Bayesian networks can perform better than the state-of-the-art machine learning models in addition to quantifying uncertainties. 

P. Thiagarajan, P. Khairnar and S. Ghosh, "Explanation and Use of Uncertainty Quantified by Bayesian Neural Network Classifiers for Breast Histopathology Images," in IEEE Transactions on Medical Imaging, vol. 41, no. 4, pp. 815-825, April 2022, doi: 10.1109/TMI.2021.3123300. Read more


P Thiagarajan, S Ghosh, "A Jensen-Shannon Divergence Based Loss Function for Bayesian Neural Networks", arXiv preprint arXiv:2209.11366

Bayesian Calibration and Uncertainty Quantification of Computational Models

Model predictions at different strain rates with experimental data and the uncertainty in predictions given as 95% confidence interval.

Computational models are mathematical representations of physical phenomena that are developed to better understand and analyze the phenomena. Most computational models have unknown parameters that cannot be measured directly from experiments. In order to make predictions using these computational models, it is necessary to calibrate these unknown parameters using observed experimental responses. In this work, a rate-dependent cohesive zone model is developed in order to predict fractures in polymeric materials. The unknown parameters in this model are calibrated using a Bayesian approach and the discrepancy between the model prediction and the experimental response is obtained. The Bayesian approach provides the posterior of the unknown model parameters. The uncertainty in the model response is then estimated by forward propagation of the posterior of the unknown parameters along with the evaluated discrepancy. This uncertainty is a measure to evaluate the confidence with which the computational model predicts the response of the actual physical phenomenon. It is important to understand this measure to efficiently use the model in any practical application.