Research

“All models are wrong, but some models are useful.”

- George Box

My research interest lies at the intersection of the following two disciplines:

Machine Learning

Scientific machine learning, Data- and Physics-driven machine learning techniques.
Probabilistic machine learning, Neural network (NN) architectures (Graph NN, Spiking NN, and Neural ODEs).
Physics-based deep generative models for uncertainty quantification and propagation.
Quantum computing and quantum machine learning algorithms.
LLMs using transformers, Autonomous Vehicles.

Scientific Computation

Inverse Problems
Multi-fidelity data and models.
Domain Decomposition Methods, High Performance Computing (HPC)
Computational Continuum Mechanics ( High speed flows, Acoustics, and Nonlinear Elasticity).
Multi-scale/multi-physics simulations.
Spectral/Finite Element methods, WENO and Discontinuous Galerkin schemes.
Fractional/Non-local PDEs

My Research Work

Extended Physics-Informed Neural Networks (XPINNs) : A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations

Highlights: XPINNs: A generalized space-time domain decomposition based framework is introduced in deep learning framework for nonlinear PDEs, which efficiently lends itself to parallelized computation thereby reduces the network training cost drastically.

Kinetic theory based multi-level finite difference WENO schemes for compressible Euler equations.

Density contours for Rayleigh-Talyor Instability using Adaptive Order Kinetic WENO schemes (WENO-AO-K1 and WENO-AO-K2)

Density contours for Kelvin-Helmholtz Instability using Adaptive Order Kinetic WENO schemes (WENO-AO-K1 and WENO-AO-K2)

Conservative Physics-Informed Neural Networks (cPINN) on discrete domains for conservation laws

X-dir velocity

Y-dir velocity

cPINN and Ghia comparison

cPINN : Solution of lid-driven cavity flow test case ( Incompressible Navier-Stokes equations, Re = 100) using cPINN method.

cPINN : (Left to right) Exact, predicted and point-wise error in the solution of 2D Burgers equation at final time 0.5.

Highlights: A conservative PINN algorithm is developed using domain decomposition approach, which can be fully parallelized. The small sub-domains are stitched together by using the conservative flux.

Ref: AD Jagtap et al., Conservative Physics-Informed Neural Networks on discrete domains for conservation laws: Applications to forward and inverse problems., Computer Methods in Applied Mechanics and Engineering, 365 (2020) 113028.

Adaptive activation functions for deep and physics-informed neural networks

Adaptive activation functions in deep and physics-informed neural networks.

1D Discontinuous function approximation using Fixed activation function.

1D Discontinuous function approximation using Adaptive activation function.

Highlight: Adaptive activations in the neural network can converge much faster than the traditional fixed activation functions.

Ref: AD Jagtap et al., Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics 404 (2020) 109136.

Physics-informed neural networks for high speed compressible flows with shock waves

1D Euler test case - Sod's shock tube reference and neural network (NN) solutions.

2D Euler oblique shock wave test case - neural network (NN) solutions.

Highlight: The PINN algorithm can capture both shock and contact waves exactly in 1D Sod's shock tube problem.

Ref. Z. Mao, AD Jagtap, GE Karniadakis, Physics-informed neural networks for high speed flows, Comput. Methods Appl. Mech. Engrg. 360 (2020) 112789.

Method of Relaxed Streamline Upwinding for Hyperbolic Conservation Laws

KKP rotating wave : Non-convex flux function

Transonic flow over a bump (Mach 0.85) : Pressure contours (top) and Mach number variation along bottom wall (bottom)

Highlight: Relaxation system based stabilized FEM method is proposed for conservation laws like Burger's equation, Euler equations and shallow water equations.

Ref: AD Jagtap, Method of Relaxed Streamline Upwinding for Hyperbolic Conservation Laws, Wave Motion, 78, 132-161, 2018.

Spectral element method for inhomogeneous sine-Gordon Equation with Impulsive Forcing

Solution of homogeneous sine Gordon equation using 6th order spectral element method on 6 x 6 elements.

Solution of sine Gordon equation for time dependent impulsive forcing

Ref: A D Jagtap, A S V Murthy, Higher Order Scheme for Two-Dimensional Inhomogeneous sine-Gordon Equation with Impulsive Forcing, Communications in Nonlinear Science and Numerical Simulation, 64, 178-197, 2018.

Spectral element method for two- and three-dimensional Cahn-Hilliard Equation

Solution of 2D Cahn-Hilliard Equation : 8 x 8 fourth order spectral element mesh

Isocontours of solution of 3D Cahn-Hilliard Equation : 5 x 5 x 5 fourth order spectral element mesh

Ref : A D Jagtap, A S V Murthy, Higher order spectral element scheme for two-and three-dimensional Cahn–Hilliard equation, International Journal of Advances in Engineering Sciences and Applied Mathematics, Vol. 10, Issue 1 , Page 79, 2018 .

L1-type smoothness indicators based WENO scheme for nonlinear degenerate parabolic equations

Barenblatt solution

Ref : S Rathan, R Kumar, A D Jagtap, L1-type smoothness indicators based WENO scheme for nonlinear degenerate parabolic equations, Applied Mathematics and Computation, 375 (2020) 125112.