Forked Physics Informed Neural Networks for Coupled Systems of Differential equations

TL;DR

This paper introduces Forked PINNs (FPINN), a novel neural network architecture designed to effectively solve coupled differential equations by stabilizing training and avoiding local optima, demonstrated on quantum dynamics models.

Contribution

The paper proposes a forked architecture for PINNs with an evolution regularization loss, improving training stability and solution accuracy for coupled systems of differential equations.

Findings

FPINN accurately models non-Markovian quantum dynamics.

FPINN outperforms standard PINNs in capturing quantum coherence revival.

The framework is broadly applicable across physics and AI domains.

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.