Quantum Dynamics via Score Matching on Bohmian Trajectories

Abstract: We solve the time-dependent Schrödinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schrödinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.

• The paper introduces a self-consistent score matching framework that recasts the TDSE as a deterministic flow on Bohmian trajectories.
• It employs neural networks and Fisher divergence minimization to accurately recover quantum potentials and manage high-dimensional, time-dependent density evolution.
• Numerical experiments on Morse chains demonstrate a four-order-of-magnitude loss reduction and 0.2% energy error, validating the method's efficiency.

Quantum Dynamics via Score Matching on Bohmian Trajectories

Introduction and Theoretical Foundation

This work presents an overview between Bohmian quantum dynamics and score-based generative modeling, formulating real-time quantum propagation as self-consistent score matching on deterministic Bohmian trajectories. The framework leverages the equivalence between the time-dependent Schrödinger equation (TDSE) and a Lagrangian ensemble of non-crossing trajectories guided by the quantum potential, $Q$, itself explicitly expressed in terms of the score function $s = \nabla \ln \rho$, where $\rho$ is the probability density. By parametrizing the score via neural networks and optimizing a global Fisher divergence functional, the dynamics become a continuous normalizing flow that is self-generated and self-regularized.

The core insight is the self-consistent variational principle: the score network $s_\theta(x, t)$ drives a deterministic flow of particle trajectories, and its associated quantum potential closes the equations of motion without explicit reference to the wavefunction phase during integration. The key computational loop ensures that the learned score converges to the true gradient of the log-density as per the dynamics it induces. The loss functional is minimized iff the flow recovers nodeless Schrödinger evolution exactly.

Figure 1: Quantum dynamics realized as a continuous normalizing flow, with non-crossing Bohmian trajectories sampling evolving densities.

Self-Consistent Learning Mechanism

The training protocol directly minimizes the Fisher divergence—the expected squared mismatch between the network score and the target score—over configuration and time, on particle ensembles generated by the current network itself. The deformation gradient $F$ (the Jacobian of flow with respect to initial conditions) crucially enables computing both the self-consistent density and the precise score target at every step, ensuring information is propagated through all derivatives required by the quantum trajectory dynamics. Learning proceeds by backpropagation through time (BPTT), supported by memory checkpointing.

Figure 2: The training loop, showing how the score network determines quantum forces, updates trajectories, and propagates loss gradients.

The proof of convergence demonstrates that, provided the loss vanishes and the flow remains nonsingular (i.e., the evolving density stays strictly positive), the learned score and quantum potential uniquely determine a nodeless Schrödinger solution, matching both observed densities and implicit phases.

Numerical Experiments: Morse Chain Quantum Vibrations

The method is applied to a chain of $d$ coupled Morse oscillators in the anharmonic regime. Utilizing up to $M = 5000$ particles for training, the network architecture includes harmonic analytical priors and time-conditioned FiLM MLPs, efficiently handling high-dimensional, time-dependent density evolution without basis-set limitations.

Learning dynamics are robustly tracked: initial instabilities (caustic formation due to insufficient quantum potential at early epochs) are regularized by masking, with the network subsequently self-correcting and the loss stabilizing. The training achieves a four-order-of-magnitude loss reduction and a final energy error at the 0.2% level relative to reference Fourier-based simulations.

Figure 3: Training convergence for the $d=4$ Morse chain, demonstrating rapid reduction of Fisher loss and mean energy error.

Post-convergence, large ensembles of independent particles propagated with the trained network replicate all relevant quantum statistics. Notably, the method preserves both mean positions and fluctuations across normal modes, correctly capturing dipole propagation, coupled-mode relaxation, and anharmonic quantum breathing, with results indistinguishable from exact numerical references.

Figure 4: Comparison of moments—mean $\langle x_i \rangle$ and width $\sigma_i$—from the learned network and exact FFT reference for all modes.

The learned score field, visualized as streamlines on reduced configuration-space slices, closely aligns with exact density gradients, both initially (Gaussian) and following complex anharmonic deformation, indicating full recovery of the self-consistent quantum potential landscape.

Figure 5: Streamlines of the learned score field overlaying exact density gradients in the 4D Morse chain (visualized for $s = \nabla \ln \rho$0, $s = \nabla \ln \rho$1 subspace).

Contextualization and Implications

The approach departs from prior trajectory-based quantum methods, which either imposed restrictive basis sets or employed pointwise fitting without a global constraint. Here, global variational optimization enabled by automatic differentiation and neural networks removes basis limitations and enforces time continuity. In contrast to tVMC and PINNs, the method requires only real-valued densities, avoids Metropolis sampling, and deterministically pushes particle ensembles based on learned quantum forces—bypassing phase and sign issues intrinsic to complex wavefunctions except at nodes.

This connection between quantum dynamics and the theory of normalizing flows unifies deterministic quantum mechanics (via Bohmian theory) with the language and toolset of modern generative models. The potential for integrating techniques such as denoising score matching, flow matching, and stochastic generative approaches (such as Deep Stochastic Mechanics or SDE-based sampling) suggests a broad landscape for further development of neural quantum dynamics, including for Fokker-Planck and more general quantum-classical systems.

For fermionic wavefunctions, which are generically nodal, the present approach must be extended—e.g., via permutation-equivariant normalizing flows seeded by Slater determinants—to address the sign structure and ensure phase quantization. Such avenues are identified as promising steps toward many-electron quantum dynamics.

Conclusion

This paper establishes a neural, global-in-time score matching framework for quantum dynamics based on Bohmian trajectories, casting the solution of the time-dependent Schrödinger equation as the self-consistent minimization of the Fisher divergence between score networks and flowing quantum densities. The method demonstrates both theoretical exactness (for nodeless states) and practical efficiency (for high-dimensional systems), aligning quantum mechanics on deterministic flows with generative diffusion and flow models. Its implications include the unification of generative learning and quantum simulation, opening paths for scalable, flexible, and potentially sign-problem-free quantum dynamics in both bosonic and fermionic systems.

[See (2604.25137) for full details and code resources.]