Quantum Physics-Informed Neural Networks for Maxwell's Equations: Circuit Design, "Black Hole" Barren Plateaus Mitigation, and GPU Acceleration

TL;DR

It is demonstrated that adding an energy conservation term to the loss stabilizes training and improves the physical fidelity of the solution in the lossless free-space case, and achieves accuracy comparable to, and even greater than, the classical PINN baseline, while using a significantly smaller number of trainable parameters.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving partial differential equations (PDEs) by embedding the governing physics into the loss function associated with a deep neural network. In this work, a Quantum PINNs (QPINN) framework is proposed to solve two-dimensional (2D) time-dependent Maxwell's equations. Our approach utilizes a parametrized quantum circuit in conjunction with the classical neural network architecture and enforces physical laws, including a global energy conservation principle, during training. A quantum simulation library, TorQ, was developed to efficiently compute circuit outputs and derivatives by leveraging GPU acceleration based on PyTorch, enabling end-to-end training of the QPINN. The method was evaluated on two 2D electromagnetic wave propagation problems: one in free space (vacuum) and the other has an added dielectric medium. Multiple quantum circuit ansätze, input scales, and an added loss term were compared in a thorough ablation study. Furthermore, recent techniques to enhance PINN convergence, including random Fourier feature embeddings and adaptive time weighting, have been incorporated. Our results demonstrate that the QPINN achieves accuracy comparable to, and even greater than, the classical PINN baseline, while using a significantly smaller number of trainable parameters. This study also shows that adding an energy conservation term to the loss stabilizes training and improves the physical fidelity of the solution in the lossless free-space case. This added term helps mitigate a new kind of barren plateau (BP) related phenomenon - ``black hole'' (BH) loss landscape for the quantum experiments in that scenario. By optimizing the quantum-circuit ansatz and embedding energy-conservation constraints, our QPINN achieves up to a 19% higher accuracy on 2D Maxwell benchmark problems compared to a classical PINN.

Figures (14)

• Figure 1: Schematic of a classical PINN architecture employed in the present work for solving 2D Maxwell's equations. The inputs $(x, y, t)$ are mapped to outputs $(E_z, H_x, H_y)$. A periodic mapping layer first enforces strict periodic boundary conditions in space and learned periodicity in time, followed by a random Fourier feature (RFF) layer that expands the inputs into a higher-dimensional sinusoidal feature space. The transformed inputs then pass through fully connected hidden layers before producing the final field outputs.
• Figure 2: Schematic of the QPINN architecture. It shares the same overall structure as the PINN in Fig. \ref{['fig:pinn_architecture']}, but a quantum circuit layer is inserted near the output. The preceding classical layer and subsequent output layer adjust dimensions for the quantum layer's 7 qubits (each qubit acting as a neuron).
• Figure 3: Effect of input-angle scalings for the $\mathrm{Rx}$ embedding and $Z$ readout. (a) General case with linear inputs $a\in[-1,1]$: with $\langle Z\rangle=\cos\theta$, $scale_{acos}$ ($\theta=\arccos a$) yields $\langle Z\rangle=a$ (identity) and $scale_{asin}$ ($\theta=\arcsin a+\tfrac{\pi}{2}$) yields $\langle Z\rangle=-a$ (sign flip). (b) Same analysis for our actual setting, where the PQC receives $\tanh$-bounded activations $a\in[-1,1]$ from the preceding layer. (c) Distribution of the scaled angles $\theta$ induced by each mapping in Eqs. \ref{['eq:scale']} for $a\sim \mathrm{Unif}[-1,1]$. (d) Probability distribution of the Pauli-Z measurement outcomes for the corresponding inputs in (c).
• Figure 4: Schematics of all ansätze used in this study. (a) Basic Entangling Layers. (b) Strongly Entangling Layers. (c) Cross-Mesh. (d) Cross-Mesh-2-Rotations. (e) Cross-Mesh-CNOT. (f) No Entanglement Ansatz.
• Figure 5: (a) Initial conditions (i.e. $t=0$) for both test cases, with the training size grid ($64^2$). Contours of the electric field $E_z$ at the final time of the wave propagation are shown for (b) vacuum at $t=1.5$ and (c) dielectric medium at $t=0.7$, as obtained from the QPINN simulations. The corresponding classical PINN and Padé scheme solutions appear visually identical; please refer to shaviner2025pinns for the classical results. The region occupied by the dielectric medium is shaded in (c).