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Deep Stochastic Mechanics
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Proposes Deep Stochastic Mechanics, which learns the current and osmotic velocities of a Nelson-style stochastic mechanics diffusion with neural networks and samples trajectories to solve the time-dependent Schrödinger equation with quadratic dimension complexity.
Problem
Simulating the time-dependent Schrödinger equation in high dimensions, where grid-based and standard neural solvers suffer from the curse of dimensionality.
Approach
Learns the osmotic and current velocities of a Markovian diffusion (stochastic mechanics) with neural networks, using gradient-of-divergence velocity equations that avoid the Laplacian, so that trajectories sampled from the diffusion concentrate computation in high-density regions and loss evaluation scales as O(d^2).
Key results
- For the 1d harmonic oscillator with zero initial phase, DSM achieves mean error 0.079 and variance error 0.019 versus PINN's 0.877 and 0.766.
- For two interacting bosons, DSM achieves mean error 0.092 vs t-VMC 0.103, and a velocity error of 7.6e-5 vs 2.9e-3 for t-VMC.
- Memory of DSM grows linearly in dimension (d=2 to 5: 17.0 to 33.5 GB), while Crank-Nicolson memory grows exponentially and cannot run for d>4 on the authors' system.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DSM instead of PINN | Harmonic oscillator d=1, S0(x)≡0 · Harmonic oscillator d=1, S0(x)≡0 | mean relative L2 error: 0.877 ± 0.263 -> 0.079 ± 0.007 | Helped |
| Use DSM instead of Gaussian sampling | Harmonic oscillator d=1, S0(x)≡0 · Harmonic oscillator d=1, S0(x)≡0 | mean relative L2 error: 0.355 ± 0.038 -> 0.079 ± 0.007 | Helped |
| Use DSM instead of PINN | Harmonic oscillator d=1, S0(x)=-5x · Harmonic oscillator d=1, S0(x)=-5x | mean relative L2 error: 2.626 ± 0.250 -> 0.268 ± 0.036 | Helped |
| Replace gradient of divergence with the Nelsonian Laplacian formulation (ablation) | DSM · Harmonic oscillator d=3, S0(x)≡0 | mean relative L2 error: 0.075 ± 0.004 vs 0.080 ± 0.015 | No clear effect |
| Replace gradient of divergence with the Nelsonian Laplacian formulation (ablation) | DSM · Harmonic oscillator d=3, S0(x)≡0 | variance relative L2 error: 0.015 ± 0.004 vs 0.016 ± 0.007 | No clear effect |
| Use DSM instead of PINN | Interacting bosons d=2 · Interacting bosons d=2 | variance relative L2 error: 1.937 ± 0.654 -> 0.055 ± 0.015 | Helped |
| Use DSM instead of t-VMC | Interacting bosons d=2 · Interacting bosons d=2 | mean relative L2 error: 0.103 ± 0.007 -> 0.092 ± 0.004 | Helped |
| Replace trajectory-based DSM sampling with Gaussian sampling (ablation) | DSM · Singular initial condition | averaged mean L2 error: 0.008 ± 0.007 -> 0.043 ± 0.042 | Hurt |
| Replace trajectory-based DSM sampling with Gaussian sampling (ablation) | DSM · Singular initial condition | averaged variance relative L2 error: 0.011 ± 0.007 -> 0.146 ± 0.013 | Hurt |
Surprising findings
- Gaussian (naive) sampling of collocation points performs substantially worse than trajectory-based sampling from the learned diffusion.
- Empirical time and memory scaling is closer to linear in d than the theoretical quadratic bound.
Limitations
- Only the linear, spinless Schrödinger equation on flat R^d with smooth potential is considered; fermionic systems are left for future work.
- Only bosonic systems are evaluated and the adaptation to low-dimensional structure is shown for one kind of structure only.
- PINN could not be run in the 3d setting due to collocation-point memory requirements.
Experimental setup
Datasets: numerical solutions of the Schrödinger equation (SciPy, qmsolve) used as ground truth
Benchmarks: 1d harmonic oscillator 3d harmonic oscillator two interacting bosons in harmonic trap
Metrics: relative L2 error of mean relative L2 error of variance relative L2 error of current velocity v relative L2 error of osmotic velocity u time per epoch memory usage training time
Baselines: PINN t-VMC Crank-Nicolson numerical solver Gaussian sampling Nelsonian DSM formulation
Builds on: Nelson stochastic mechanics PINN time-dependent variational Monte Carlo Euler-Maruyama
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