arXiv subjects

Hardeep Bassi

Publications and source records attributed to Hardeep Bassi.

7 recordsLinked to original sources

arXiv2025

RELift: Learned Coarse-to-Fine Propagators for Time-Dependent PDEs with Applications to Electron Dynamics

We present RELift (Restrict, Evolve, Lift), a two-phase learning framework that couples coarse-grid numerical solvers with neural operators to super-resolve and forecast fine-grid dynamics for time-dependent partial differential equations (PDEs). In Phase 1, RELift learns a super-resolution operator that maps the solution on a coarse grid to a fine grid. In Phase 2, this learned operator is composed with a coarse-grid numerical integrator to construct an effective fine-grid propagator for the governing equation. We benchmark RELift on three canonical two-dimensional PDEs of increasing dynamical complexity -- the heat equation, the wave equation, and the incompressible Navier--Stokes equations -- and we further demonstrate its performance on a kinetic electron dynamics case study via the 1D1V Vlasov--Poisson system. Across all examples, RELift delivers high-fidelity super-resolution (Phase 1) and accurate long-horizon rollouts (Phase 2), outperforming standard super-resolution and neural operator baselines in both field-level error metrics and physics-relevant diagnostics. Finally, we provide error analysis of the effective fine-grid propagator, characterizing how approximation errors accumulate over time and explaining the observed numerical stability of the RELift framework.

math.NA↗

arXiv2025

From noisy observables to accurate ground state energies: a quantum classical signal subspace approach with denoising

We propose a hybrid quantum-classical algorithm for ground state energy (GSE) estimation that remains robust to highly noisy data and exhibits low sensitivity to hyperparameter tuning. Our approach -- Fourier Denoising Observable Dynamic Mode Decomposition (FDODMD) -- combines Fourier-based denoising thresholding to suppress spurious noise modes with observable dynamic mode decomposition (ODMD), a quantum-classical signal subspace method. By applying ODMD to an ensemble of denoised time-domain trajectories, FDODMD reliably estimates the system's eigenfrequencies. We also provide an error analysis of FDODMD. Numerical experiments on molecular systems demonstrate that FDODMD achieves convergence in high-noise regimes inaccessible to baseline methods under a limited quantum computational budget, while accelerating spectral estimation in intermediate-noise regimes. Importantly, this performance gain is entirely classical, requiring no additional quantum overhead and significantly reducing overall quantum resource demands.

quant-ph↗

arXiv2025

Transformer-based operator learning framework for self-energy in strongly correlated systems

We introduce $\Sigma$-Attention, a Transformer-based operator-learning framework to address a key computational challenge in correlated materials. Our approach utilizes an Encoder-Only Transformer as an ansatz to approximate the self-energy operator of strongly correlated electronic systems. By creating a batched dataset that combines results from three complementary approaches: many-body perturbation theory, strong-coupling expansion, and exact diagonalization, each effective in specific parameter regimes, $\Sigma$-Attention is applied to learn a universal approximation for the self-energy operator that is valid across all regimes. This hybrid strategy leverages the strengths of existing methods while relying on the transformer's ability to generalize beyond individual limitations. More importantly, the scalability of the Transformer architecture allows the learned self-energy to be extended to systems with larger sizes, leading to much improved computational scaling. Using the 1D Hubbard model, we demonstrate that $\Sigma$-Attention can accurately predict the Matsubara Green's function and hence effectively captures the Mott transition at finite temperature. Our framework offers a promising and scalable pathway for studying strongly correlated systems with many possible generalizations.

cond-mat.str-el↗

arXiv2024

Nonlinear Optimal Control of Electron Dynamics within Hartree-Fock Theory

Consider the problem of determining the optimal applied electric field to drive a molecule from an initial state to a desired target state. For even moderately sized molecules, solving this problem directly using the exact equations of motion -- the time-dependent Schr\"odinger equation (TDSE) -- is numerically intractable. We present a solution of this problem within time-dependent Hartree-Fock (TDHF) theory, a mean field approximation of the TDSE. Optimality is defined in terms of minimizing the total control effort while maximizing the overlap between desired and achieved target states. We frame this problem as an optimization problem constrained by the nonlinear TDHF equations; we solve it using trust region optimization with gradients computed via a custom-built adjoint state method. For three molecular systems, we show that with very small neural network parametrizations of the control, our method yields solutions that achieve desired targets within acceptable constraints and tolerances.

math.OC↗

arXiv2024

Incorporating Memory into Propagation of 1-Electron Reduced Density Matrices

For any linear system with unreduced dynamics governed by invertible propagators, we derive a closed, time-delayed, linear system for a reduced-dimensional quantity of interest. This method does not target dimensionality reduction: rather, this method helps shed light on the memory-dependence of $1$-electron reduced density matrices in time-dependent configuration interaction (TDCI), a scheme to solve for the correlated dynamics of electrons in molecules. Though time-dependent density functional theory has established that the $1$-electron reduced density possesses memory-dependence, the precise nature of this memory-dependence has not been understood. We derive a symmetry/constraint-preserving method to propagate reduced TDCI electron density matrices. In numerical tests on two model systems ($\text{H}_2$ and $\text{HeH}^+$), we show that with sufficiently large time-delay (or memory-dependence), our method propagates reduced TDCI density matrices with high quantitative accuracy. We study the dependence of our results on time step and basis set. To implement our method, we derive the $4$-index tensor that relates reduced and full TDCI density matrices. Our derivation applies to any TDCI system, regardless of basis set, number of electrons, or choice of Slater determinants in the wave function.

math.DS↗

arXiv2023

Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the need to perform a numerical integration in each numerical time evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can be reduced to $O(n_T)$ from $O(n_T^2)$ if a $n_T$-step trajectory is to be computed. We illustrate the efficiency and robustness of this LSTM-RNN-based numerical IDE solver with a model problem. Additionally, we highlight the generalizability of the learned integral operator by applying it to IDEs driven by different external forces. As a practical application, we show how this methodology can effectively solve the Dyson's equation for quantum many-body systems.

cs.LG↗

arXiv2020

Learning to predict synchronization of coupled oscillators on randomly generated graphs

Suppose we are given a system of coupled oscillators on an unknown graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? Even with a known underlying graph structure, this is an important yet analytically intractable question in general. In this work, we take an alternative approach to the synchronization prediction problem by viewing it as a classification problem based on the fact that any given system will eventually synchronize or converge to a non-synchronizing limit cycle. By only using some basic statistics of the underlying graphs such as edge density and diameter, our method can achieve perfect accuracy when there is a significant difference in the topology of the underlying graphs between the synchronizing and the non-synchronizing examples. However, in the problem setting where these graph statistics cannot distinguish the two classes very well (e.g., when the graphs are generated from the same random graph model), we find that pairing a few iterations of the initial dynamics along with the graph statistics as the input to our classification algorithms can lead to significant improvement in accuracy; far exceeding what is known by the classical oscillator theory. More surprisingly, we find that in almost all such settings, dropping out the basic graph statistics and training our algorithms with only initial dynamics achieves nearly the same accuracy. We demonstrate our method on three models of continuous and discrete coupled oscillators -- the Kuramoto model, Firefly Cellular Automata, and Greenberg-Hastings model. Finally, we also propose an "ensemble prediction" algorithm that successfully scales our method to large graphs by training on dynamics observed from multiple random subgraphs.

math.DS↗