Forked Physics Informed Neural Networks for Coupled Systems of Differential equations

TL;DR

The proposed FPINN architecture offers a general and effective framework for solving coupled systems of equations, which arise across a broad spectrum from classical physics to modern artificial intelligence, including applications in multi-body rotational dynamics, multi-asset portfolio optimization, chemical reaction kinetics, and deep representation learning.

Abstract

Solving coupled systems of differential equations (DEs) is a central problem across scientific computing. While Physics Informed Neural Networks (PINNs) offer a promising, mesh-free approach, their standard architectures struggle with the multi-objective optimization conflicts and local optima traps inherent in coupled problems. To address the first issue, we propose a Forked PINN (FPINN) framework designed for coupled systems of DEs. FPINN employs a shared base network with independent branches, isolating gradient pathways to stabilize training. We demonstrate the effectiveness of FPINN in simulating non-Markovian open quantum dynamics governed by coupled DEs, where multi-objective conflicts and local optima traps often cause evolutionary stagnation. To overcome this second challenge, we incorporate an evolution regularization loss that guides the model away from trivial solutions and ensures physically meaningful evolution. We demonstrate the effectiveness of FPINN in simulating non-Markovian open quantum dynamics governed by coupled DEs, where multi-objective conflicts and local optima traps often cause evolutionary stagnation. For the spin-boson and XXZ models, FPINN accurately captures hallmark non-Markovian features, such as quantum coherence revival and information backflow, significantly outperforming standard PINNs. The proposed FPINN architecture offers a general and effective framework for solving coupled systems of equations, which arise across a broad spectrum from classical physics to modern artificial intelligence, including applications in multi-body rotational dynamics, multi-asset portfolio optimization, chemical reaction kinetics, and deep representation learning.

Figures (11)

• Figure 1: (a) FPINNs architecture for simulating the dynamics of the auxiliary operators $\overline{O}(t)$ and $\overline{Q}(t)$. The red circle represents the input layer with $t$ as the input. The green dashed part represents the shared layer of FPINN, which consists of three fully connected layers used to learn the common features of different operators. The shared layer is connected to two branched layers (the yellow dashed part), each composed of a middle layer and an output layer, used to learn and output the unique dynamical feature vectors $N_o(t)$ and $N_q(t)$ for different operators. (b) PINNs architecture for simulating the dynamics of density matrix $\rho$. The output layers of PINN and FPINN are the same, with the hidden layer consisting of three fully connected layers, which output the eigenvector $N_{\rho}(t)$ of the density matrix from the output layer.
• Figure 2: The evolution of the spin-boson model is simulated by RK4 and FPINN. The sampling time points $t_f = 201$, the maximum evolution time $T_{\text{tot}} = 6$, and the environmental parameters are $\Gamma = 0.1$, $\gamma = 0.3$, $T = 20$. (a), (b), and (c) show the results for $\overline{O}_{12}$, $\overline{Q}_{21}$, and $\langle \sigma_z \rangle$, respectively.
• Figure 3: (a) and (b) show the evolution of the auxiliary operators $\overline{O}$ and $\overline{Q}$ for the spin-boson model simulated by RK4, FPINN, UPINN, and SPINN, respectively. $\Gamma = 0.1$, $\gamma = 0.5$, $T = 20$.
• Figure 4: The evolution of the XXZ model is simulated by RK4 and FPINN. The sampling time points $t_f = 401$, the maximum evolution time $T_{\text{tot}} = 6$, and the environmental parameters are $\Gamma = 0.1$, $\gamma = 0.4$, $T = 20$. (a) and (b) show the evolution of the auxiliary operators $O$ and $Q$, respectively, while (c) and (d) show the evolution of quantum coherence and quantum entanglement for different initial states.
• Figure 5: Evolution of the operators $\overline{O}$ and $\overline{Q}$ for the spin‑boson model simulated by FPINN and RK4 for different $\gamma$. Panels (a,c,e) show $\overline{O}(t)$, panels (b,d,f) show $\overline{Q}(t)$. The first row has parameters $\gamma = 0.3$, the second row $\gamma = 0.5$, and the third row $\gamma = 0.7$ ($\Gamma = 0.1$ and $T = 20$). The sampling time points $t_f = 201$, the maximum evolution time $T_{\text{tot}} = 6$.