Research Work

Abstract: Graph-based representations for samples of computational mechanics-related datasets can prove instrumental when dealing with problems like irregular domains or molecular structures of materials, etc. To effectively analyze and process such datasets, deep learning offers Graph Neural Networks (GNNs) that utilize techniques like message-passing within their architecture. The issue, however, is that as the individual graph scales and/ or GNN architecture becomes increasingly complex, the increased energy budget of the overall deep learning model makes it unsustainable and restricts its applications in applications like edge computing. To overcome this, we propose in this paper Variable Spiking Graph Neural Networks (VS-GNNs) and their hybrid variants, collectively termed VS-GNN architectures, that utilize Variable Spiking Neurons (VSNs) within their architecture to promote sparse communication and hence reduce the overall energy budget. VSNs, while promoting sparse event-driven computations, also perform well for regression tasks, which are often encountered in computational mechanics applications and are the main target of this paper. Three examples dealing with the prediction of mechanical properties of materials based on their microscale/ mesoscale structures are shown to test the performance of the proposed VS-GNNs architectures in regression tasks. We have compared the performance of VS-GNN architectures with the performance of vanilla GNNs, GNNs utilizing leaky integrate and fire neurons, and GNNs utilizing recurrent leaky integrate and fire neurons. The results produced show that VS-GNN architectures perform well for regression tasks, all while promoting sparse communication and, hence, energy efficiency.

Abstract: In this paper, we propose a novel data-driven operator learning framework referred to as the Randomized Prior Wavelet Neural Operator (RP-WNO). The proposed RP-WNO is an extension of the recently proposed wavelet neural operator, which boasts excellent generalizing capabilities but cannot estimate the uncertainty associated with its predictions. RP-WNO, unlike the vanilla WNO, comes with inherent uncertainty quantification module and hence, is expected to be extremely useful for scientists and engineers alike. RP-WNO utilizes randomized prior networks, which can account for prior information and is easier to implement for large, complex deep-learning architectures than its Bayesian counterpart. Four examples have been solved to test the proposed framework, and the results produced advocate favorably for the efficacy of the proposed framework.

Abstract: Energy-efficient deep learning algorithms are essential for a sustainable future and feasible edge computing setups. Spiking neural networks (SNNs), inspired from neuroscience, are a positive step in the direction of achieving the required energy efficiency. However, in a bid to lower the energy requirements, accuracy is marginally sacrificed. Hence, it becomes important to quantify the uncertainties in such models originating from limited and noisy data, surrogate gradients, and non-convex optimization encountered during training. In response to this challenge, we introduce the Conformalized Randomized Prior Operator (CRP-O) framework that leverages Randomized Prior (RP) networks and Split Conformal Prediction (SCP) to quantify uncertainties in both conventional and spiking neural operators. To further enable zero-shot super-resolution, we propose an extension incorporating Gaussian Process Regression. This enhanced super-resolution-enabled CRP-O framework is integrated with the recently developed Variable Spiking Wavelet Neural Operator (VSWNO). To test the performance of the obtained calibrated uncertainty bounds, we discuss four different benchmark examples covering both one-dimensional and two-dimensional partial differential equations. Results demonstrate that the uncertainty bounds produced by the conformalized RP-VSWNO significantly enhance the uncertainty estimates compared to vanilla RP-VSWNO, Quantile WNO (Q-WNO), and Conformalized Quantile WNO (CQ-WNO). These findings underscore the potential of the proposed approach for practical applications.

Abstract: We propose, in this paper, a Variable Spiking Wavelet Neural Operator (VS-WNO), which aims to bridge the gap between theoretical and practical implementation of Artificial Intelligence (AI) algorithms for mechanics applications. With recent developments like the introduction of neural operators, AI's potential for being used in mechanics applications has increased significantly. However, AI's immense energy and resource requirements are a hurdle in its practical field use case. The proposed VS-WNO is based on the principles of spiking neural networks, which have shown promise in reducing the energy requirements of the neural networks. This makes possible the use of such algorithms in edge computing. The proposed VS-WNO utilizes variable spiking neurons, which promote sparse communication, thus conserving energy, and its use is further supported by its ability to tackle regression tasks, often faced in the field of mechanics. Various examples dealing with partial differential equations, like Burger's equation, Allen Cahn's equation, and Darcy's equation, have been shown. Comparisons have been shown against wavelet neural operator utilizing leaky integrate and fire neurons (direct and encoded inputs) and vanilla wavelet neural operator utilizing artificial neurons. The results produced illustrate the ability of the proposed VS-WNO to converge to ground truth while promoting sparse communication.

Abstract: Assumptions and approximations made while analyzing any physical system induce modeling uncertainty, which, if left unchecked, can result in the erroneous analysis of the system under consideration. Additionally, the discrepancy in the exact knowledge of system parameters can further result in deviation from the ground truth. This paper explores Physics-integrated Variational Auto-Encoder (PVAE) to account for modeling and parametric uncertainties in partially known nonlinear dynamical systems. The PVAE under consideration has three main parts: encoder, latent space, and decoder. The complete PVAE architecture is employed during the training stage of the machine learning model, while only the decoder is used to make the final predictions. The encoder determines the correct parameter values for the known part of the model (in the form of a known ODE). The decoder is augmented with an ODE solver that solves the known part of the system and the estimated discrepancy together to reconstruct the measurements. To test the efficacy of the PVAE architecture, three case studies are carried out, each presenting unique challenges. The probability density functions obtained for the various systems’ responses demonstrate the efficacy of the PVAE architecture. Furthermore, reliability analysis has been carried out, and the results produced have been compared against those obtained from a multi-layered, densely connected forward neural network.

Abstract: Neural network based data-driven operator learning schemes have shown tremendous potential in computational mechanics. DeepONet is one such neural network architecture which has gained widespread appreciation owing to its excellent prediction capabilities. Having said that, being set in a deterministic framework exposes DeepONet architecture to the risk of overfitting, poor generalization and in its unaltered form, it is incapable of quantifying the uncertainties associated with its predictions. To address these challenges, we propose a novel Bayesian operator learning framework referred to as the Variational Bayes DeepONet (VB-DeepONet). VB-DeepONet is rooted in Bayesian statistics and hence, (a) is less prone to overfitting as compared to its deterministic counterpart, (b) has better generalization, and (c) yields predictive uncertainty which is instrumental when decision making is concerned. VB-DeepONet exploits variational inference and hence has the capacity to take into account high dimensional posterior distributions while keeping the associated computational cost reasonable. Different examples covering mechanics problems like diffusion reaction, gravity pendulum, advection diffusion have been considered to illustrate the performance of the proposed VB-DeepONet and comparisons have been drawn against DeepONet set in deterministic framework, Proper Orthogonal Decomposition based Gaussian Process and DenseED. The results obtained illustrate the efficacy of the proposed approach in solving uncertainty quantification problems.

Abstract: Time dependent reliability analysis and uncertainty quantification of structural system subjected to stochastic forcing function is a challenging endeavour as it necessitates considerable computational time. We investigate the efficacy of recently proposed DeepONet in solving time dependent reliability analysis and uncertainty quantification of systems subjected to stochastic loading. Unlike conventional machine learning and deep learning algorithms, DeepONet is an operator network and learns a function to function mapping and hence, is ideally suited to propagate the uncertainty from the stochastic forcing function to the output responses. We use DeepONet to build a surrogate model for the dynamical system under consideration. Multiple case studies, involving both toy and benchmark problems, have been conducted to examine the efficacy of DeepONet in time dependent reliability analysis and uncertainty quantification of linear and nonlinear dynamical systems. Comparisons have also been drawn with Recurrent Neural Network results and with results obtained from Proper Orthogonal Decomposition based Gaussian process. The results obtained indicate that the DeepONet architecture is accurate as well as efficient. Moreover, DeepONet posses zero shot learning capabilities and hence, a trained model easily generalizes to unseen and new environment with no further training.

Abstract: For real-life nonlinear systems, the exact form of nonlinearity is often not known and the known governing equations are often based on certain assumptions and approximations. Such representation introduce model-form error into the system. In this paper, we propose a novel gray-box modeling approach that not only identifies the model-form error but also utilizes it to improve the predictive capability of the known but approximate governing equation. The primary idea is to treat the unknown model-form error as a residual force and estimate it using dual Bayesian filter based joint input-state estimation algorithms. For improving the predictive capability of the underlying physics, we first use machine learning algorithm to learn a mapping between the estimated state and the input (model-form error) and then introduce it into the governing equation as an additional term. This helps in improving the predictive capability of the governing physics and allows the model to generalize to unseen environment. Although in theory, any machine learning algorithm can be used within the proposed framework, we use Gaussian process in this work. To test the performance of proposed framework, case studies discussing four different dynamical systems are discussed; results for which indicate that the framework is applicable to a wide variety of systems and can produce reliable estimates of original system’s states. Apart from this, the algorithm has also been tested for a case where the data has been taken from an experimental setup (Silver box dataset). The results produced further showcase the efficacy of the proposed framework.

Abstract: The potential of digital twin technology is immense, specifically in the infrastructure, aerospace, and automotive sector. However, practical implementation of this technology is not at an expected speed, specifically because of lack of application-specific details. In this paper, we propose a novel digital twin framework for stochastic nonlinear multi-degree of freedom (MDOF) dynamical systems. The proposed digital twin has four modules — (a) a physics-based nominal model, (b) a data collection module, (c) algorithm for real-time update of the digital twin and (d) module for predicting future state. The modules for real-time update and prediction are based on the so-called gray-box modeling approach, and utilizes both physics based and data driven frameworks; this enables the proposed digital twin to generalize and predict future responses. The gray box modeling framework used within the digital twin is developed by coupling Bayesian filtering and machine learning algorithm. Although, the proposed digital twin can be used with any machine learning regression algorithm, we have used Gaussian process in this study. Performance of the proposed approach is illustrated using two examples. Results obtained indicate the applicability and excellent performance of the proposed digital twin framework.

Abstract:  Reliability analysis of engineering systems under uncertainty poses significant computational challenges, particularly for problems involving high-dimensional stochastic inputs, nonlinear system responses, and multiphysics couplings. Traditional surrogate modeling approaches often incur high energy consumption, which severely limits their scalability and deployability in resource-constrained environments. We introduce NeuroPOL, the first neuroscience-inspired physics-informed operator learning framework for reliability analysis. NeuroPOL incorporates Variable Spiking Neurons into a physics-informed operator architecture, replacing continuous activations with event-driven spiking dynamics. This innovation promotes sparse communication, significantly reduces computational load, and enables an energy-efficient surrogate model. The proposed framework lowers both computational and power demands, supporting real-time reliability assessment and deployment on edge devices and digital twins. By embedding governing physical laws into operator learning, NeuroPOL builds physics-consistent surrogates capable of accurate uncertainty propagation and efficient failure probability estimation, even for high-dimensional problems. We evaluate NeuroPOL on five canonical benchmarks, the Burgers equation, Nagumo equation, two-dimensional Poisson equation, two-dimensional Darcy equation, and incompressible Navier-Stokes equation with energy coupling. Results show that NeuroPOL achieves reliability measures comparable to standard physics-informed operators, while introducing significant communication sparsity, enabling scalable, distributed, and energy-efficient deployment.

Abstract:  We introduce NeuroPINNs, a neuroscience-inspired extension of Physics-Informed Neural Networks (PINNs) that incorporates biologically motivated spiking neuron models to achieve energy-efficient PDE solving. Unlike conventional PINNs, which rely on continuously firing activations and therefore incur high computational and energy costs, NeuroPINNs leverage Variable Spiking Neurons (VSNs) to enable sparse, event-driven communication. This makes them particularly well-suited for deployment on neuromorphic hardware and for scenarios with constrained computational resources, such as embedded and edge devices. A central challenge, however, lies in reconciling the discontinuous dynamics of spiking neurons with the smooth residual-based loss formulation required in PINNs. Direct smoothing introduces systematic biases, leading to inaccurate PDE learning. To overcome this, we employ a novel stochastic projection method inspired from upscaled theory that faithfully captures spiking behavior while maintaining compatibility with gradient-based optimization. Standard surrogate backpropagation is used for parameter updates, ensuring computational tractability. We demonstrate the effectiveness of NeuroPINNs on four representative PDE problems across both regular and irregular domains. Furthermore, application of NeuroPINN for linear elastic micromechnics in three dimensions was also explored. Results show that NeuroPINNs achieve high accuracy while substantially reducing communication and energy demands, marking a step toward scalable, neuromorphic-ready scientific machine learning.

Abstract:  Robust uncertainty quantification (UQ) remains a critical barrier to the safe deployment of deep learning in real-time virtual sensing, particularly in high-stakes domains where sparse, noisy, or non-collocated sensor data are the norm. We introduce the Conformalized Monte Carlo Operator (CMCO), a framework that transforms neural operator-based virtual sensing with calibrated, distribution-free prediction intervals. By unifying Monte Carlo dropout with split conformal prediction in a single DeepONet architecture, CMCO achieves spatially resolved uncertainty estimates without retraining, ensembling, or custom loss design. Our method addresses a longstanding challenge: how to endow operator learning with efficient and reliable UQ across heterogeneous domains. Through rigorous evaluation on three distinct applications: turbulent flow, elastoplastic deformation, and global cosmic radiation dose estimation-CMCO consistently attains near-nominal empirical coverage, even in settings with strong spatial gradients and proxy-based sensing. This breakthrough offers a general-purpose, plug-and-play UQ solution for neural operators, unlocking real-time, trustworthy inference in digital twins, sensor fusion, and safety-critical monitoring. By bridging theory and deployment with minimal computational overhead, CMCO establishes a new foundation for scalable, generalizable, and uncertainty-aware scientific machine learning.

Abstract:  Redundant information transfer in a neural network can increase the complexity of the deep learning model, thus increasing its power consumption. We introduce in this paper a novel spiking neuron, termed Variable Spiking Neuron (VSN), which can reduce the redundant firing using lessons from biological neuron inspired Leaky Integrate and Fire Spiking Neurons (LIF-SN). The proposed VSN blends LIF-SN and artificial neurons. It garners the advantage of intermittent firing from the LIF-SN and utilizes the advantage of continuous activation from the artificial neuron. This property of the proposed VSN makes it suitable for regression tasks, which is a weak point for the vanilla spiking neurons, all while keeping the energy budget low. The proposed VSN is tested against both classification and regression tasks. The results produced advocate favorably towards the efficacy of the proposed spiking neuron, particularly for regression tasks.

Abstract:  We explore the concept of digital twins for dynamical systems. In particular, we discuss purely physics-based and gray-box model-based digital twins in this chapter. While physics-based models allow better generalization, a purely physics-based digital twin is often not robust because of noise in the data. On the other hand, gray-box modeling-based digital twin allows seamless fusion of data and physics. One of the primary challenges associated with digital twins is robustness. In its journey toward autonomy, a digital twin should be probabilistic, allowing for better decision-making. We discuss how coupling the Gaussian process with a physics-based solver will enable us to develop a robust and probabilistic digital twin framework. Possible extensions to stochastic systems are also discussed in this chapter. A digital twin is supposed to track a systems/asset’s evolution throughout its lifetime. However, the governing physics of dynamical systems often evolves with time; hence, a digital twin needs to track changes in governing physics. We discuss how sparse Bayesian learning can be used to track changes in governing physics and how it can aid in the development of digital twin technology. Aspects related to data quality and sampling rate are analyzed to understand the practicality of the digital twin technology.