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.
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.
Physics-informed neural networks (PINNs) provide a pathway to reunify the simulation and control of quantum systems, in which these two tasks are typically decoupled in traditional strategies. However, most work remains confined to Markovian environments. When applied to non-Markovian systems, standard PINN architectures fail to converge reliably due to multi-objective optimization conflicts arising from the coupled differential equations. To address this fundamental limitation, we extend our previously proposed forked PINN (FPINN) by incorporating a dedicated control branch. By decoupling the optimization objectives at the gradient level via selective gradient flow, our method turns a previously intractable multi-task optimization into a well-conditioned one, allowing simulation and control to be optimized jointly without compromise. Numerical simulations on a two-qubit Heisenberg XXX model confirm that our framework faithfully reproduces the features of non-Markovian dynamics, including decoherence and information backflow. Taking a state-preparation task on the same model as an example, our FPINN achieves higher fidelity than gradient ascent pulse engineering, chopped random basis, and standard PINNs, with the advantage becoming more pronounced as the environment becomes more dissipative and more Markovian. The generated pulses are also noticeably smoother, which is advantageous for experimental implementation. Our framework thus provides a unified, end-to-end differentiable paradigm for simulation and control of open quantum systems, with potential implications for quantum computing, simulation, and control.
Physics-informed neural networks (PINNs) and hybrid quantum-classical extensions provide a promising framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. In this work, we study embedding strategies for trainable embedding quantum physics-informed neural networks (TE-QPINNs) in the context of nonlinear reaction-diffusion (RD) systems. We introduce an extended TE-QPINN (x-TE-QPINN) architecture that supports both classical and fully quantum embeddings, enabling a controlled comparison between feedforward neural network-based feature maps and parameterized quantum circuit embeddings. The first architecture is the classical embedding feed-forward neural network-based TE-QPINN (FNN-TE-QPINN), while the latter variant is a purely quantum one, referred to as quantum embedding neural network-based TE-QPINN (QNN-TE-QPINN). The proposed framework employs hardware-efficient variational quantum circuits and species-specific readout operators to approximate coupled multi-field dynamics while enforcing governing equations, boundary conditions, and initial conditions through a physics-informed loss function. By isolating the embedding mechanism while keeping the variational ansatz, loss formulation, and optimization procedure fixed, we analyze the impact of embedding design on gradient structure, parameter scaling, and quantum resource requirements. Numerical experiments on one- and two-dimensional RD equations demonstrate that quantum embeddings can replace classical embeddings without degradation in solution accuracy and, in certain regimes, exhibit improved optimization behavior compared to classical PINNs and hybrid quantum models with fixed embeddings. These results provide architectural insight into hybrid quantum PDE solvers and inform the design of resource-efficient quantum physics-informed learning methods.
Physics-informed neural networks (PINNs) represent a methodology that integrates physical equations into neural networks. By incorporating the residuals of physical equations as part of the loss function, PINNs enable the network to learn data features while simultaneously satisfying the constraints imposed by physical laws. The underlying principle of PINNs involves using optimization algorithms to iteratively update the network parameters until the value of a specified physics-informed loss function decreases to an acceptable level, thereby driving the network towards the solution of the differential equations. This approach is not merely a tool but serves as a bridge integrating scientific computing with artificial intelligence, offering novel paradigms for modeling and predicting complex systems. In this paper, we propose a non-Hermitian shortcut to adiabaticity (STA) scheme based on physics-informed neural networks. We utilize the physics-informed neural networks to solve parameterized differential equations, employing the neural networks as an approximating function for the quantum adiabatic evolution process. This provides an interpretable and end-to-end framework (termed STAPINNs) for control-field optimization, distinct from traditional gradient-based approaches like gradient ascent pulse engineering (GRAPE), which require pre-defined pulse parameterization. The parameterized differential equations, along with various physical constraints, are incorporated into the network's loss function. Training the networks allows them to fit the quantum system's evolution process and obtain the driving control function for population inversion. Compared to conventional STA techniques, our approach introduces machine learning into non-Hermitian quantum systems, enabling highly robust and high-fidelity population transfer. Numerical simulations demonstrate that the proposed method significantly outperforms traditional STA protocols under dissipative conditions, achieving fidelities above 0.99 even in the presence of substantial decoherence. Moreover, the framework demonstrates strong generalization across wide parameter ranges and is inherently scalable to multi-level and many-body systems due to the mesh-free, high-dimensional handling capability of PINNs. Neural networks possess strong computational capabilities suitable for generating driving control functions in complex systems, and are equally applicable to non-Hermitian STA techniques. The STAPINNs framework not only enhances control flexibility but also provides a powerful tool for optimizing quantum operations in open and noisy systems.
Abstract We demonstrate that system-agnostic physics-informed neural networks (PINNs) can efficiently learn the dynamical solution of quantum master equations, as compared to problem-specific numerical solvers, with respect to accuracy, stability of training convergence, robustness towards bath parameters or initial conditions, and equivalent convergence times for the non-Markovian evolution of two-level quantum systems. We study cases of quantum dot systems driven by external electromagnetic fields and coupled to an acoustic phonon environment, and analyze two problem setups, the evolution of Bloch vectorized coupled differential equations obtained from a Lindblad master equation and the evolution of a non-Markovian master equation obtained from the Nakajima–Zwanzig formulation. We implement a fully connected PINN architecture to solve the two problems. We also use the same methodology to solve the inverse problem of estimation of bath parameters in a supervised fashion, but with a very limited number of noisy samples.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
In this work, we introduce the Quantum-Classical Hybrid Physics-Informed Neural Network with Multiplicative and Additive Couplings (QPINN-MAC): a novel hybrid architecture that integrates the framework of Physics-Informed Neural Networks (PINNs) with that of Quantum Neural Networks (QNNs). Specifically, we prove that through strategic couplings between classical and quantum components, the QPINN-MAC retains the universal approximation property, ensuring its theoretical capacity to represent complex solutions of ordinary differential equations (ODEs). Simultaneously, we demonstrate that the hybrid QPINN-MAC architecture actively mitigates the barren plateau problem, regions in parameter space where cost-function gradients decay exponentially with circuit depth, a fundamental obstacle in QNNs that hinders optimization during training. Furthermore, we prove that these couplings prevent gradient collapse, ensuring trainability even in high-dimensional regimes. Thus, our results establish a new pathway for constructing quantum-classical hybrid models with theoretical convergence guarantees, which are essential for the practical application of QPINNs.