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arXiv:2609.04527

Learning Memory-Dependent Neural Network Correlation Potentials for Accurate Electron Dynamics

Harish S. Bhat, Christine M. Isborn

Published Sep 3, 2026• 0 citations
arXivHTMLPDF

Abstract

Time-dependent density functional theory, though exact in theory, is in practice applied in an adiabatic approximation using exchange-correlation functionals with only local temporal dependence. Simultaneously, the exact correlation potential formally depends on, among other quantities, the time-history of electron densities. Here we develop a framework to learn neural network models of correlation functionals that feature explicit memory-dependence. Our framework includes two different approaches, both linked through their use of adjoint-based optimization. One approach decouples the learning of the functional from inversion to find ground truth values of the correlation potential. The other approach learns the functional directly without requiring inversion. We apply these methods to modeling the electron dynamics of two-electron systems in two spatial dimensions. In both cases, our methods yield correlation functionals with low test set propagation error, outperforming standard local density and generalized gradient approximation functionals by one to two orders of magnitude. Overall, our framework points at one strategy to move beyond the adiabatic approximation and develop memory-dependent correlation functionals that yield accurate propagation for excited state and/or non-equilibrium dynamics.

DOI
10.48550/arxiv.2609.04527

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Time-dependent density functional theory, though exact in theory, is in practice applied in an adiabatic approximation using exchange-correlation functionals with only local temporal dependence. Simultaneously, the exact correlation potential formally depends on, among other quantities, the time-history of electron densities. Here we develop a framework to learn neural network models of correlation functionals that feature explicit memory-dependence. Our framework includes two different approaches, both linked through their use of adjoint-based optimization. One approach decouples the learning of the functional from inversion to find ground truth values of the correlation potential. The other approach learns the functional directly without requiring inversion. We apply these methods to modeling the electron dynamics of two-electron systems in two spatial dimensions. In both cases, our methods yield correlation functionals with low test set propagation error, outperforming standard local density and generalized gradient approximation functionals by one to two orders of magnitude. Overall, our framework points at one strategy to move beyond the adiabatic approximation and develop memory-dependent correlation functionals that yield accurate propagation for excited state and/or non-equilibrium dynamics.
Abstract The exact exchange-correlation potential v xc in time-dependent density functional theory is a causal functional of the full density history, a dependence that widely used adiabatic approximations discard. We ask whether this history dependence can be approximated by a learnable finite-dimensional memory state while retaining a Hamiltonian-consistent description of the Kohn–Sham phase-space dynamics. To this end, we propose a Three-Head neural architecture consisting of a Hamiltonian dynamics head that induces a symplectic vector field from a parameterized scalar energy, a direct vector field head that supplies a flexible correction pathway, and a memory XC potential head that reconstructs v xc from density history through a recurrent memory module with cross-window carry. We evaluate the model on two physically distinct benchmarks. On the one-dimensional two-electron electron–hydrogen scattering model, it attains the lowest density reconstruction error among all baselines over the full propagation window, by the largest margin during the pre-collision approach phase, and is the only model that recovers the peak-and-step structure of v xc in the subsequent strongly nonadiabatic scattering window. On a one-dimensional helium Rabi oscillation model under strong-field driving, it reduces the correlation-potential reconstruction error by roughly an order of magnitude relative to all adiabatic baselines at every driven phase of both the Rabi and optical cycles, including the adiabatically exact ground-state limit, which shows that the residual error along the adiabatic axis is controlled by the missing density history rather than by the accuracy of the time-local map. Ablation studies show that structured dynamics and memory-based potential learning are mutually supportive rather than separable.
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A framework for variational time-dependent current-density-functional theory beyond linear response and adiabatic approximations is given. The method allows incorporation of quantum memory into exchange-correlation functionals which respect Vignale’s “Galilean invariance” (GI) principle (Phys. Rev. Lett. 74, 3233 (1995)). The free variables of the theory are the electron velocity-field and density. Gauge-invariant time-dependent (TD) Kohn-Sham equations are derived. The action functionals allow memory effects and are expressed in terms Lagrangian quantities, to ensure GI. General closed-form expressions for the corresponding scalar and vector potentials are derived and applied within a specific functional, based on the homogeneous electron gas (HEG). The scalar potential depends on the future velocity field while the vector potential depends on both future and past velocity fields. These future dependencies naturally arise when non-local temporal effects are allowed and do not violate physical causality.
We propose a framework to learn the time-dependent Hartree-Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Though the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange-correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60×60×60×6060 \times 60 \times 60 \times 6060×60×60×60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.
We propose a framework to learn the time-dependent Hartree–Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Although the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange–correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60 × 60 × 60 × 60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.
Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.