Debojyoti Ghosh

Shrirang Abhyankar, Jed Brown, Emil M. Constantinescu et al.

High-quality ordinary differential equation (ODE) solver libraries have a long history, going back to the 1970s. Over the past several years we have implemented, on top of the PETSc linear and nonlinear solver package, a new general-purpose, extensive, extensible library for solving ODEs and differential algebraic equations (DAEs). Package includes support for both forward and adjoint sensitivities that can be easily utilized by the TAO optimization package, which is also part of PETSc. The ODE/DAE integrator library strives to be highly scalable but also to deliver high efficiency for modest-sized problems. The library includes explicit solvers, implicit solvers, and a collection of implicit-explicit solvers, all with a common user interface and runtime selection of solver types, adaptive error control, and monitoring of solution progress. The library also offers enormous flexibility in selection of nonlinear and linear solvers, including the entire suite of PETSc iterative solvers, as well as several parallel direct solvers.

Christophe Bonneville, Youngsoo Choi, Debojyoti Ghosh et al.

Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the latent space dynamics. By interpolating and solving the ODE system in the reduced latent space, fast and accurate ROM predictions can be made by feeding the predicted latent space dynamics into the decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework that relies on Gaussian process (GP) for latent space ODE interpolations. Using GPs offers two significant advantages. First, it enables the quantification of uncertainty over the ROM predictions. Second, leveraging this prediction uncertainty allows for efficient adaptive training through a greedy selection of additional training data points. This approach does not require prior knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently non-intrusive and can be applied to problems without a known PDE or its residual. We demonstrate the effectiveness of our approach on the Burgers equation, Vlasov equation for plasma physics, and a rising thermal bubble problem. Our proposed method achieves between 200 and 100,000 times speed-up, with up to 7% relative error.

Debojyoti Ghosh, Emil M. Constantinescu

This paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time-scale is significantly faster than the advective scale, yet it is typically not relevant to atmospheric and weather phenomena. The acoustic and advective components of the hyperbolic flux are separated in the characteristic space. High-order, conservative additive Runge-Kutta methods are applied to the partitioned equations so that the acoustic component is integrated in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale. Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared with that of explicit time integration.

Christophe Bonneville, Xiaolong He, April Tran et al.

Numerical solvers of partial differential equations (PDEs) have been widely employed for simulating physical systems. However, the computational cost remains a major bottleneck in various scientific and engineering applications, which has motivated the development of reduced-order models (ROMs). Recently, machine-learning-based ROMs have gained significant popularity and are promising for addressing some limitations of traditional ROM methods, especially for advection dominated systems. In this chapter, we focus on a particular framework known as Latent Space Dynamics Identification (LaSDI), which transforms the high-fidelity data, governed by a PDE, to simpler and low-dimensional latent-space data, governed by ordinary differential equations (ODEs). These ODEs can be learned and subsequently interpolated to make ROM predictions. Each building block of LaSDI can be easily modulated depending on the application, which makes the LaSDI framework highly flexible. In particular, we present strategies to enforce the laws of thermodynamics into LaSDI models (tLaSDI), enhance robustness in the presence of noise through the weak form (WLaSDI), select high-fidelity training data efficiently through active learning (gLaSDI, GPLaSDI), and quantify the ROM prediction uncertainty through Gaussian processes (GPLaSDI). We demonstrate the performance of different LaSDI approaches on Burgers equation, a non-linear heat conduction problem, and a plasma physics problem, showing that LaSDI algorithms can achieve relative errors of less than a few percent and up to thousands of times speed-ups.

Debojyoti Ghosh, Adrijit Goswami

Accurate waste classification is vital for achieving sustainable waste management and reducing the environmental footprint of urbanization. Misclassification of recyclable materials contributes to landfill accumulation, inefficient recycling, and increased greenhouse gas emissions. To address these issues, this study introduces HybridSOMSpikeNet, a hybrid deep learning framework that integrates convolutional feature extraction, differentiable self-organization, and spiking-inspired temporal processing to enable intelligent and energy-efficient waste classification. The proposed model employs a pre-trained ResNet-152 backbone to extract deep spatial representations, followed by a Differentiable Soft Self-Organizing Map (Soft-SOM) that enhances topological clustering and interpretability. A spiking neural head accumulates temporal activations over discrete time steps, improving robustness and generalization. Trained on a ten-class waste dataset, HybridSOMSpikeNet achieved a test accuracy of 97.39%, outperforming several state-of-the-art architectures while maintaining a lightweight computational profile suitable for real-world deployment. Beyond its technical innovations, the framework provides tangible environmental benefits. By enabling precise and automated waste segregation, it supports higher recycling efficiency, reduces contamination in recyclable streams, and minimizes the ecological and operational costs of waste processing. The approach aligns with global sustainability priorities, particularly the United Nations Sustainable Development Goals (SDG 11 and SDG 12), by contributing to cleaner cities, circular economy initiatives, and intelligent environmental management systems.

Debojyoti Ghosh, Soumya K Ghosh, Adrijit Goswami

Efficient and accurate classification of waste and industrial surface defects is essential for ensuring sustainable waste management and maintaining high standards in quality control. This paper introduces the Neuroplastic Modular Classifier, a novel hybrid architecture designed for robust and adaptive image classification in dynamic environments. The model combines a ResNet-50 backbone for localized feature extraction with a Vision Transformer (ViT) to capture global semantic context. Additionally, FAISS-based similarity retrieval is incorporated to provide a memory-like reference to previously encountered data, enriching the model's feature space. A key innovation of our architecture is the neuroplastic modular design composed of expandable, learnable blocks that dynamically grow during training when performance plateaus. Inspired by biological learning systems, this mechanism allows the model to adapt to data complexity over time, improving generalization. Beyond garbage classification, we validate the model on the Kolektor Surface Defect Dataset 2 (KolektorSDD2), which involves industrial defect detection on metal surfaces. Experimental results across domains show that the proposed architecture outperforms traditional static models in both accuracy and adaptability. The Neuroplastic Modular Classifier offers a scalable, high-performance solution for real-world image classification, with strong applicability in both environmental and industrial domains.