Research Projects
Our work draws from computational science, applied mathematics, and engineering and runs in three directions. The first is the use of classical and physics-informed machine learning methods for inverse problems – recovering what cannot be directly measured from what can – with thermal and fluid systems as our primary ground, though the questions travel further. The second is Bayesian inverse problems and uncertainty quantification – characterising not just the most likely answer but the full range of answers consistent with the data, through amortized variational inference and generative models that make such estimates fast enough for practical use. The third is the development of structure-preserving high-order numerical schemes for PDEs – methods that respect the deep mathematical structure of the underlying physics.
Inverse Problems and PDE-Constrained Optimization in Thermo-Fluid Systems: Classical and Data-Driven Approaches
Active Our group develops and applies classical and physics-informed machine learning methods to recover unknown parameters, boundary conditions, and source terms from indirect measurements -- with the goal of understanding, monitoring, and predicting the behaviour of thermo-fluid and related engineering systems. On the data-driven side, we work with PINNs, physics-informed neural operators (PINO) and related techniques. On the classical side, adjoint-based methods remain a primary tool when the problem demands it. Parameter estimation and real-time monitoring are the central targets, and the longer ambition threading through all of this is the development of digital twins for complex engineering systems.
Although our primary focus is on thermal and fluid systems, though we remain genuinely open to problems wherever the mathematics is interesting and the physics is hard.
Methods & Tools
- Physics-informed neural networks (PINNs)
- Physics-informed neural operators (PINO)
- Discrete adjoint-based sensitivity and gradient computation
- Tikhonov and total-variation regularisation
Students & Staff
- Pradipan Maitra -- MS student
- Abhishek Srivastava -- Postdoctoral fellow
- Jariful Hassan — Research Staff
Collaborators
- Prof. Anand TNC -- Professor, Mechanical Engineering, IIT Palakkad
- Dr. Sourav Sarkar -- Assistant Professor, Mechanical Engineering, Jadavpur University
Publications
- PINN-based Estimation of Convective Heat Transfer in Jet Impingement Cooling (2026)
- Generalized PINN-Based Inverse Estimation of Spatially Varying Heat Transfer Coefficients with Transfer Learning (2026, submitted)
Bayesian Inverse Problems and Uncertainty Quantification
ActiveMany inverse problems are ill-posed: the measurements are consistent with many possible answers. Rather than return a single best estimate, we treat the unknown as a random variable and characterise its full posterior distribution, so that every reconstruction comes with a statement of how much it can be trusted. Classical sampling methods such as MCMC are accurate but too expensive for repeated or real-time use. We work on amortized Bayesian inference, in which a generative model is trained once, offline, to map measurements directly to posterior distributions. We want to understand when these methods work, when they fail, and why -- how training objectives, priors, and model design shape the answers they give, and how far their uncertainty estimates can actually be trusted. Just as much, we want to put them to use on real inverse problems in physics and engineering.
Methods & Tools
- Bayesian formulation of inverse problems
- Amortized variational inference
- Normalising flows and variational autoencoders (VAE)
- Posterior calibration and uncertainty quantification
Students & Staff
- Abhishek Srivastava -- Postdoctoral fellow
Publications
High-Order Structure-Preserving Schemes for Computational Electrodynamics and MHD
Past / ContinuingA central interest of our group is the development of numerical schemes for partial differential equations that faithfully mimic the structure of the underlying physical laws -- not merely in accuracy, but in the preservation of fundamental mathematical and physical properties.
One direction where we have worked extensively is the preservation of curl-type constraints -- $\nabla \cdot \mathbf{B} = 0$ and $\nabla \cdot \mathbf{D} = \rho -- intrinsic to Maxwell's equations and ideal MHD. Violating these constraints numerically leads to unphysical solutions and long-time instability. We develop high-order Discontinuous Galerkin (DG) and Flux Reconstruction (FR) schemes that preserve these constraints globally -- not just in a weak or cell-averaged sense -- using multidimensional Riemann solvers and carefully constructed numerical fluxes. To extend these ideas further, we developed two-derivative Runge–Kutta time integrators and generalised multidimensional Riemann solvers as core building blocks.
A closely related interest is the development of low-dissipation, low-dispersion schemes for computational electrodynamics (CED) and related areas such as aeroacoustics. Achieving this requires careful attention at every level of the discretisation -- high-order methods to control dissipation, and structure-preserving time integrators such as symplectic schemes to control dispersion and maintain long-time fidelity. For realistic simulations, adaptive mesh refinement and the combination of artificial boundary conditions with high-order methods are equally important building blocks.
Methods & Tools
- Discontinuous Galerkin (DG) and Flux Reconstruction (FR) schemes
- Globally divergence-free and curl-free evolution
- Multidimensional Riemann problem solvers (classical and generalised)
- Two-derivative Runge-Kutta time stepping
- von Neumann and dispersion–dissipation analysis
Collaborators
- Prof. Praveen Chandrashekar -- TIFR-Centre for Applicable Mathematics, Bengaluru
- Prof. Dinshaw S. Balsara -- University of Notre Dame, Department of Physics
- Dr. Sudip K. Garain -- IISER Kolkata, Department of Physics
Selected Papers
Deep Learning for Geophysical Inverse Problems
Past / ContinuingGeosteering--subsurface characterization using borehole electromagnetic resistivity measurements from logging-while-drilling (LWD) instruments -- is one of the major techniques employed in subsurface characterization, natural resource extraction, geothermal exploration, and CO₂ sequestration. Classical regularization-based inversion methods, while reliable, are too slow for the near-real-time decisions that geosteering demands. We adopted and applied deep neural network (DNN) based inversion methods that learn a direct mapping from electromagnetic measurements to subsurface properties offline, reducing evaluation time from minutes to fractions of a second. Building on an existing two-step training strategy combining forward model approximation with data-misfit minimization, we developed a greedy instrument design algorithm that selects a minimal set of measurement types sufficient for accurate subsurface characterization. Ongoing directions include recasting the inversion as a statistical inference problem -- using variational autoencoders and generative models to estimate both subsurface properties and their associated uncertainties, without the computational cost of traditional MCMC methods.
Methods & Tools
- 1D Hankel transform-based forward modelling of borehole electromagnetic measurements
- Deep neural network-based inversion
- Two-step training with data-misfit minimisation
- Greedy algorithm for optimal instrument design
- Variational autoencoders and generative models for statistical inversion
- Uncertainty quantification via Bayesian inference
Students & Staff
- Manas K. Sinha -- M.Tech (Computing and Mathematics)
Collaborators
- Prof. David Pardo -- Basque Centre for Applied Mathematics (BCAM), University of Basque Country
Selected Papers
Real-Time MRI Simulation
PastThe Bloch equations govern the evolution of nuclear magnetization and form the mathematical foundation of magnetic resonance imaging (MRI). My doctoral work focused on numerical simulation of the Bloch equations for flowing objects -- with the goal of enabling quantitative estimation of fluid flow velocities and flow rates from MR magnitude signals acquired in standard clinical sequences. The governing equation, cast as a vector-valued reactive transport PDE, was solved using an operator splitting approach -- separating the transport and reaction terms and treating each with appropriate numerical methods. High-resolution finite volume methods (FVM) handled magnetization transport, while the magnetic resonance phenomena were resolved through further splitting. The simulator was validated against in vitro experiments and applied to Couette, laminar, and pulsatile flows. Subsequent work established the well-posedness of the modified Bloch model for flowing objects and explored accuracy improvements using a Discontinuous Galerkin FEM (DGFEM) scheme.
Methods & Tools
- Bloch equation modelling for flowing objects
- Operator splitting for reactive transport PDEs
- High-resolution finite volume methods (FVM) and Discontinuous Galerkin FEM (DGFEM)
- CUDA-enabled GPU parallelisation
- Well-posedness analysis of modified Bloch model