Research

Topics

1. Bayesian SciML in materials science

   Desc.

   Autonomous materials discovery has intensified the need to extract structural relationships between candidate materials and their physical properties from experimental data—a challenge naturally cast as symbolic regression. While methods like Deep Symbolic Regression (DSR), Equation Learner Network (EQL), QLattice, and AI‑Feynman excel in low‑noise settings, they lack a comprehensive, model‑based approach for quantifying uncertainty under realistic experimental noise. My work fills this gap by bridging the realms of scientific machine learning and statistical modeling. We develop a hierarchical Bayesian symbolic regression framework that not only learns interpretable scientific expressions via symbolic trees but also delivers principled uncertainty quantification and rigorous model selection within a unified probabilistic paradigm.

2. SciML for seismic wave modeling

   Desc.

   Physics-informed machine learning (PIML) has transformed the way for obtaining numerical solutions of partial differential equations (PDEs) arising out of physics-based models across various scientific disciplines—from chemistry and earth sciences to geophysics. In particular, high-fidelity wave equation simulations are essential for National Lab applications such as subsurface imaging, ground-based exploration, and non-destructive ultrasound imaging. However, these class of simulations remain prohibitively expensive even when performed on supercomputing platforms. During the summer 2025 graduate internship at Los Alamos National Laboratory, my work focussed on developing a novel scalable machine learning model, combining the strengths of Fourier neural operators (FNOs) and physics-informed neural networks (PINNs) to solve seismic wave equations in the complex media.

3. Approximate Bayesian methods

   Desc.

   Variational inference (VI, popularly known as variational Bayes (VB)) has emerged as a scalable alternative to Markov chain Monte Carlo (MCMC) for approximating complex Bayesian posteriors, but a fundamental challenge remains in rigorously characterizing its statistical behavior. A part of my research at Texas A&M Statistics aims at developing the theoretical and algorithmic foundations of tangent-transform VI for a broad and flexible class of probability models under the umbrella of strongly super-
Gaussian likelihoods. In particular, we provide general conditions for obtaining optimal risk bounds for point estimates under this regime, analyze the stability and convergence of the resulting algorithm, and demonstrate its effectiveness through applications to real-world scientific problems.

4. Bayesian optimization

   Desc.

   Bayesian optimization (BO) tackles expensive black‑box optimization problems by placing a prior—typically a Gaussian process (GP) prior—on the objective and sequentially choosing evaluations to balance exploration and exploitation. While classical methods like GP-UCB and expected improvement (EI) rely on domain discretization and covering-number bounds, my research on BO develops fractional GP-Thompson sampling (TS), which uses a fractional GP to mitigate posterior under-dispersion. By invoking first-order posterior contraction theory, we obtain frequentist regret guarantees without any discretization. This streamlined analysis matches state-of-the-art rates and paves the way for robust TS variants in complex settings.

Ongoing

1. SPINWAVE

   SPINWAVE: Scalable Physics-Informed Neural Operator for Seismic WAVE Modeling

   Somjit Roy, Kai Gao and Ting Chen.

   In preparation.
2. mHierBOSSS

   Multi-Property Materials Discovery using multivariate HierBOSSS

   Somjit Roy, Pritam Dey, Bani K. Mallick, Debdeep Pati and Raymundo Arróyave.

   In preparation.
3. GP-TS

   Frequentist Regret Analysis of Fractional Gaussian Process Thompson Sampling

   Somjit Roy, Prateek Jaiswal, Anirban Bhattacharya, Debdeep Pati and Bani K. Mallick.

   In preparation.

2025

1. HierBOSSS

   Hierarchical Bayesian Operator-induced Symbolic Regression Trees for Structural Learning of Scientific Expressions

   Somjit Roy, Pritam Dey, Bani K. Mallick and Debdeep Pati.

   Submitted (Under review).

   Abs

   The advent of Scientific Machine Learning (SciML) has heralded a transformative era in scientific discovery, driving progress across a wide array of applications in major scientific domains. Central to this progress is the task of uncovering underlying scientific laws and expressions from experimental data, through symbolic regression. However, existing methods rely on data-intensive machine learning architectures trained on low-noise conditions, along with ad hoc and heuristic strategies that lack a principled framework for uncertainty quantification. In this article, we propose a novel hierarchical Bayesian model for symbolic regression consisting of an operator-induced sum-of-symbolic trees that nonparametrically model the structure of scientific expressions, coupled with regularized prior specification. This coherent probabilistic framework enables full posterior inference using an efficient Markov chain Monte Carlo algorithm, facilitating both accurate prediction and structural learning of interpretable symbolic expressions. We develop a model selection criterion based on the joint marginal posterior distribution of the symbolic tree structures given the data, guided by the Occam's window principle. Additionally, we assess the structural proximity between the learned expressions and the true symbolic relationships using a tailored distance metric. Robust performance of our proposed methodology is presented with a bake-off against state-of-the-art competing symbolic regression modules in a simulated example, a suite of popular canonical Feynman equations, and the single-atom catalysis dataset.

2. TAVIE

   A Generalized Tangent Approximation Framework for Strongly Super-Gaussian Likelihoods

   Somjit Roy, Pritam Dey, Debdeep Pati and Bani K. Mallick.

   arXiv:2504.05431.

   Abs

   Tangent approximation form a popular class of variational inference (VI) techniques for Bayesian analysis in intractable non-conjugate models. It is based on the principle of convex duality to construct a minorant of the marginal likelihood, making the problem tractable. Despite its extensive applications, a general methodology for tangent approximation encompassing a large class of likelihoods beyond logit models with provable optimality guarantees is still elusive. In this article, we propose a general Tangent Approximation based Variational InferencE (TAVIE) framework for strongly super-Gaussian (SSG) likelihood functions which includes a broad class of flexible probability models. Specifically, TAVIE obtains a quadratic lower bound of the corresponding log-likelihood, thus inducing conjugacy with Gaussian priors over the model parameters. Under mild assumptions on the data-generating process, we demonstrate the optimality of our proposed methodology in the fractional likelihood setup. Furthermore, we illustrate the empirical performance of TAVIE through extensive simulations and an application on the U.S. 2000 Census real data.

2024

1. JSTP

   Almost Perfect Mutually Unbiased Bases that are Sparse

   Ajeet Kumar, Subhamoy Maitra and Somjit Roy.

   Journal of Statistical Theory and Practice. .

   Abs Journal

   Selected ideas of statistical designs are exploited in this paper in constructions related to Mutually Unbiased Bases (MUBs). In dimension $d$, MUBs are a collection of orthonormal bases over $\mathbb{C}^d$ such that for any two vectors $v_1, v_2$ belonging to different bases, the dot or scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$. The upper bound on the number of such bases is $d+1$. Construction methods to achieve this bound are known for cases when $d$ is some power of prime. The situation is more restrictive in other cases and also when we consider the results over real rather than complex. Thus, certain relaxations of this model are considered in literature and consequently Approximate MUBs (AMUB) are studied. This enables one to construct potentially large number of such objects for $\mathbb{C}^d$ as well as in $\mathbb{R}^d$. In this regard, we propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of inner product $|\braket{v_1|v_2}|$ to be two-valued, one being 0 and the other $ \leq \frac{1+\mathcal{O}(d^{-\lambda})}{\sqrt{d}}$, such that $\lambda > 0$ and the numerator $1 + \mathcal{O}(d^{-\lambda}) \leq 2$. Each such vector constructed, has an important feature that large number of its components are zero and the non-zero components are of equal magnitude. Our techniques are based on combinatorial structures related to Resolvable Block Designs (RBDs), that are used extensively in statistical designs. We show that for several composite dimensions $d$, one can construct $\mathcal{O}(\sqrt{d})$ many APMUBs, in which cases the number of MUBs are significantly small. To be specific, this result works for $d$ of the form $(q-e)(q+f), \ q, e, f \in \mathbb{N}$, with the conditions $0 \leq f \leq e$ for constant $e, f$ and $q$ some power of prime. We also show that such APMUBs provide sets of Bi-angular vectors which are of the order of $\mathcal{O}(d^{\frac{3}{2}})$ in numbers, having high angular distances among them. Finally, as the MUBs are equivalent to a set of Hadamard matrices, we show that the APMUBs are so with the set of Weighing matrices.

2022

1. CSCML 2022

   A Heuristic Framework to Search for Approximate Mutually Unbiased Bases

   Sreejit Chaudhury, Ajeet Kumar, Subhamoy Maitra, Somjit Roy and Sourav Sen Gupta.

   In Cyber Security, Cryptology, and Machine Learning 2022. .

   Abs DOI

   Mutually Unbiased Bases (MUBs) have varied applications in quantum information. However, obtaining the optimal number of MUBs is a challenging problem for different dimensions. The problem has received serious attention for several decades and still number of questions are unsolved in this domain. As optimal number of MUBs may not always be available for different composite dimensions, Approximate MUBs (AMUBs) received serious attention in recent time. In this paper, we present a heuristic to obtain AMUBs with significantly good parameters. Given a non-prime dimension d, we note the closest prime {\$}{\$}d' > d{\$}{\$}d{\textasciiacutex}>dand form {\$}{\$}d'+1{\$}{\$}d{\textasciiacutex}+1MUBs through the existing methods. Then our proposed idea is (i) to apply basis reduction techniques (that are well studied in Machine Learning literature) in obtaining the initial solutions, and finally (ii) to exploit the steepest ascent kind of search to achieve further improved results. The efficacy of our technique is shown through construction of AMUBs in dimensions {\$}{\$}d = 6, 10, 46{\$}{\$}d=6,10,46from {\$}{\$}d' = 7, 11{\$}{\$}d{\textasciiacutex}=7,11and 47 respectively. Our technique provides a novel framework in construction of AMUBs that can be refined in a case-specific manner. From a more generic view, this approach considers approximately solving a challenging (where efficient deterministic algorithms are not known) mathematical problem in discrete domain through state-of-the-art heuristic ideas.