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Physics-informed LSTM–attention for predicting three-dimensional Brownian motion in coupled quadratic potentials
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DOI：10.1016/j.physa.2026.131800.png)
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Abstract
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We investigate the stochastic dynamics of Brownian particles in a three-dimensional coupled quadratic potential and propose LAP, an autoregressive model combining LSTM, attention, and discrete-time physics-informed constraints. Unlike continuous-time PINNs based on automatic differentiation, LAP enforces physical consistency on sampled autoregressive trajectories through a zero-mean finite-difference Langevin residual and fluctuation–dissipation-based covariance matching. Euler–Maruyama trajectories are used as numerical references, and an unconstrained LSTM–attention model, LA, serves as the baseline under the same dataset and training protocol. Results under the present experimental setup indicate that LAP reduces average displacement error, stepwise prediction error, and velocity deviation relative to LA. LAP also better reproduces the mean-squared displacement and potential-energy evolution of the reference trajectories, while showing a comparable velocity-autocorrelation decay trend. Parameter analysis shows that diagonal mass and damping terms mainly control inertial and dissipative time scales, whereas off-diagonal couplings more strongly affect trajectory geometry and turning behavior. These results suggest that LAP provides a viable framework for physically consistent trajectory prediction and parameter-effect analysis in three-dimensional quadratic potentials.
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