Simulating non-Markovian open quantum dynamics by exploiting physics-informed neural network

TL;DR

The proposed PINN-DQME method employs time-encoded neural networks within a time-domain decomposition strategy to represent the evolution governed by the dissipaton-embedded quantum master equation (DQME).

Abstract

This work integrates the physics-informed neural network (PINN) approach into the neural quantum state framework to simulate open quantum system dynamics, to circumvent the computationally expensive time-dependent variational principle required in conventional variational methods. The proposed PINN-DQME method employs time-encoded neural networks within a time-domain decomposition strategy to represent the evolution governed by the dissipaton-embedded quantum master equation (DQME). We implement and validate this approach in the single-impurity Anderson model, benchmarking the PINN-DQME results against the numerically exact hierarchical equations of motion. The PINN-DQME method demonstrates high accuracy in simulating quantum dissipative dynamics at high temperatures, where non-Markovian effects are weak. However, for strongly non-Markovian dynamics at low temperatures, it encounters challenges with error accumulation during time propagation, highlighting an area for future refinement in applying PINNs to complex quantum dynamical settings.

Figures (6)

• Figure 1: Schematic of the fermionic DQME theory, mapping the original OQS (left) to a dissipaton-embedded system (right). Red and blue bars represent the $N_{_{\rm S}}$ system fermion energy levels and $N_{_{\rm E}}$ memory-carrying dissipaton levels, respectively. Broadening of blue bars indicates each dissipaton's decay rate (inverse lifetime).
• Figure 2: Schematic of the neural network representing $\rho_{\rm pre}(\vec{n},\vec{n}';\vec{m};t)$. The input layer consists of three parts: $\{n_1,\cdots, n_{N_{_{\rm S}}}\}$ and $\{n'_1, \cdots, n'_{N_{_{\rm S}}}\}$ comprise the system part with $N_{_{\rm S}}$ being the number of the system fermion levels, $\{m_1,\cdots, m_{N_{_{\rm E}}} \}$ denote the environmental indeces with $N_{_{\rm E}}$ being the number of memory-carrying dissipaton levels, and $\{f_1(t),\cdots,f_{N_{_{\rm T}}}(t)\}$ denote a group of functions of time. Here, $n_i$ and $m_j$ represent the occupation number on $i$th system level and $j$th dissipaton level, respectively; which take the value of $0$ or $1$.
• Figure 3: (a) The evolution of the electric current flowing into the impurity from the right reservoir at different temperatures. Inset: schematic of quantum dissipative dynamics for an impurity coupled to left (L) and right (R) reservoirs with chemical potentials $\mu_{\rm L}$ and $\mu_{\rm R}$. (b) The evolution of the occupation number of the up spin electron in the impurity at two different temperatures. The purple and red dots represent the decomposition points $\{t_p\}$ for the time domain. The results of PINN-DQME (solid lines) are benchmarked against the reference values (dashed lines) obtained by the HEOM method. For the case of $k_{\rm B}T=0.3\,\Gamma$, the PINN-DQME calculation is terminated at $t_{\rm end} = 0.82\,\Gamma^{-1}$, as the minimization of loss function becomes increasingly demanding; see the main text for details. The MLP hyperparameters are chosen as $K=4$ and $N_{\rm h}=35$. For the case of $k_{_{\rm B}}T=3.0\,\Gamma$, the mapping functions $f_1(t)=t$ and $f_2(t)=f_3(t)=0$ are used in all subdomains. For the case of $k_{_{\rm B}}T=0.3\,\Gamma$, the mapping functions $f_1(t)=t$, $f_2(t)=t^2$, and $f_3(t)=t^3$ are used in the first two subdomains, and $f_1(t)=t$, $f_2(t)=t^{1.5}$, and $f_3(t)=t^{0.5}/(t+0.015\Gamma^{-1})$ are used in the later subdomains. System energetic parameters are chosen as (in units of $\Gamma$): $\epsilon_0 = U_0/2 = 2$, $\Delta\epsilon = -7$, and $\Delta U = 6$.
• Figure 4: Comparison of electric current for two loss functions for the case of $k_{\rm B}T=0.3\,\Gamma$. The main panel shows the time evolution of the electric current obtained with the original loss function (blue line) and the modified loss function (orange line); see the main text for details. Purple and red dots mark the domain‑decomposition points $t_p$. The inset provides a zoomed‑in view of the third subdomain $[0.3\,\Gamma^{-1}, 0.4\,\Gamma^{-1}]$, where the difference between the two loss functions becomes more apparent.
• Figure 5: Staged training process with adaptive residual-point sampling in the first subdomain $[0, 0.228\, \Gamma^{-1}]$ for the case of $k_{\rm B}T=0.3\,\Gamma$. (a) Evolution of the population $n_\uparrow$ during training. Solid lines show the PINN-DQME output at different stages, corresponding to successively reduced average residual-point spacings $\Delta\tau$. The dots on each curve mark the residual points used in that stage. The dashed black line represents reference data. (b) Corresponding decay of the training loss (log scale) for each stage. The colored curves track the loss progression for the $\Delta\tau$ values shown in (a), demonstrating stable convergence throughout the adaptive refinement.