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International Conference on Information Communication and Applications (ICICA)

A Comparative Analysis of Adaptive and Scheduled Dynamic Loss Weighting Strategies in Quantum Physics-Informed Neural Networks (QPINNs) for Solving the 1D TimeIndependent Schrodinger Equation

John Edoh Onuh, Ming Jiang

Published Sep 14, 2025• 1 citation
Publisher

Abstract

In recent years, Quantum Computing has revolutionized the field of physics by enabling researchers to tackle complex problems that were previously unsolvable. The ability to simulate and analyze phenomena at a quantum level has led to significant breakthroughs in our understanding of the universe. In this article, we describe and discuss how quantum computers are being used for cutting-edge research in quantum physics. The time-independent Schrödinger equation is a cornerstone of quantum mechanics, yet its solution for complex systems is a formidable computational challenge. Quantum Physics-Informed Neural Networks (QPINNs) have emerged as a promising paradigm, leveraging Quantum Neural Networks (QNNs) as a wavefunction ansatz within a physics-informed machine learning framework. A critical and often overlooked aspect of training these models is the management of the multi-term loss function, which balances adherence to the governing Partial Differential Equation (PDE) with physical constraints such as normalization. This paper presents a comprehensive implementation and comparative analysis of two distinct Dynamic Loss Weighting (DLW) strategies for training QPINNs to solve the 1D time-independent Schrödinger equation. The first strategy is an adaptive, gradient-based method that dynamically balances loss components based on their real-time impact on model parameters. The second is a pre-scheduled annealing method that follows a curriculum-like approach, prioritizing different physical constraints at different stages of training. We apply these methods to two benchmark systems: the Quantum Harmonic Oscillator (QHO) and the Finite Square Well. Our results demonstrate that the adaptive, gradient-based DLW, when paired with a physic informed wavefunction ansatz for the QHO, achieves remarkable accuracy, converging to the exact ground state energy of \(E{\rm{\ }} = {\rm{\ }}0.5\). Conversely, the scheduled annealing strategy applied to Square Well, using a more generic ansatz, converges to a plausible energy but with a higher residual loss. This comparative analysis reveals a crucial insight: the effectiveness of a DLW strategy is deeply intertwined with the physical-informedness of the underlying wavefunction ansatz. This suggests that a synergistic co-design philosophy, which considers the interplay between the quantum model's architecture and the adaptive training algorithm, is essential for developing robust and accurate QPINN-based solvers for quantum systems.

DOI
10.1145/3743158.3783855

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In recent years, Quantum Computing has revolutionized the field of physics by enabling researchers to tackle complex problems that were previously unsolvable. The ability to simulate and analyze phenomena at a quantum level has led to significant breakthroughs in our understanding of the universe. In this article, we describe and discuss how quantum computers are being used for cutting-edge research in quantum physics. The time-independent Schrödinger equation is a cornerstone of quantum mechanics, yet its solution for complex systems is a formidable computational challenge. Quantum Physics-Informed Neural Networks (QPINNs) have emerged as a promising paradigm, leveraging Quantum Neural Networks (QNNs) as a wavefunction ansatz within a physics-informed machine learning framework. A critical and often overlooked aspect of training these models is the management of the multi-term loss function, which balances adherence to the governing Partial Differential Equation (PDE) with physical constraints such as normalization. This paper presents a comprehensive implementation and comparative analysis of two distinct Dynamic Loss Weighting (DLW) strategies for training QPINNs to solve the 1D time-independent Schrödinger equation. The first strategy is an adaptive, gradient-based method that dynamically balances loss components based on their real-time impact on model parameters. The second is a pre-scheduled annealing method that follows a curriculum-like approach, prioritizing different physical constraints at different stages of training. We apply these methods to two benchmark systems: the Quantum Harmonic Oscillator (QHO) and the Finite Square Well. Our results demonstrate that the adaptive, gradient-based DLW, when paired with a physic informed wavefunction ansatz for the QHO, achieves remarkable accuracy, converging to the exact ground state energy of (E{\rm{\ }} = {\rm{\ }}0.5). Conversely, the scheduled annealing strategy applied to Square Well, using a more generic ansatz, converges to a plausible energy but with a higher residual loss. This comparative analysis reveals a crucial insight: the effectiveness of a DLW strategy is deeply intertwined with the physical-informedness of the underlying wavefunction ansatz. This suggests that a synergistic co-design philosophy, which considers the interplay between the quantum model's architecture and the adaptive training algorithm, is essential for developing robust and accurate QPINN-based solvers for quantum systems.
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 Physics-Informed Neural Network (QPINN) framework is proposed to solve two-dimensional (2D) time-dependent Maxwell’s equations. Our approach utilizes a parametrized quantum circuit (PQC) 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 - Tensor Operations for Research in Quantum systems , 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 demonstrates 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%19\%19% higher accuracy on 2D Maxwell benchmark problems compared to a classical PINN.
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.
The Schrodinger equation in one spatial dimension admits a small set of exactly solvable potentials that serve as natural proving grounds for any new eigenvalue solver. We formulate Physics-Informed Neural Networks (PINNs) and Physics-Informed Quantum Neural Networks (PIQNNs) for the time-independent Schrodinger equation and apply them to three of these benchmarks: the harmonic oscillator, the infinite square well, and the finite square well. In each case a composite loss encodes the differential-equation residual, the normalization condition, the boundary behavior, and the orthogonality between eigenstates, so that the trial wave function is driven toward a genuine eigenfunction without supervised data. The eigenvalues and wave functions returned by both methods are compared against the exact spectra and against three classical references: the matrix Numerov method, the finite difference method, and the shooting method. For the smooth oscillator the two neural solvers reproduce the lowest four eigenvalues to parts per million, while for the square wells they recover the analytic levels with comparable fidelity even where the potential is discontinuous. The quantum circuit, built as a layered angle-embedding ansatz with strongly entangling blocks, converges more reliably than its classical counterpart on the higher excited states, where the loss landscape of the classical network becomes harder to navigate.
Quantum neural networks have emerged as powerful models for approximating nonlinear functions. Yet their use in solving integro-differential equations (IDEs) and fractional integro-partial differential equations (FIPDEs), which involve inherently nonlocal operators, remains unexplored. This work introduces a quantum physics-informed neural network (QPINN) framework that combines a quantum neural network with the governing equations of general nonlinear IDEs and FIPDEs. The proposed quantum network uses an affine feature map and variational quantum circuits to produce trial solutions with explicit trigonometric structure. We prove a quantitative L2(μ)L^{2}(μ)L2(μ) universal approximation theorem for this architecture, achieving a convergence rate of O(n−1/2)\mathcal{O}(n^{-1/2})O(n−1/2). This extends classical Fourier approximation theory to quantum circuits for physics-informed learning. We propose two QPINN variants: the numerical-quadrature QPINN (N-QPINN), which handles nonlocal integrals and fractional operators via high-order numerical quadrature while computing local derivatives through automatic differentiation of quantum trial solutions; and the auxiliary-function QPINN (A-QPINN), which eliminates numerical quadrature by introducing auxiliary variables that reformulate each integro-differential equation as an equivalent coupled system of partial differential equations, enabling a multi-output quantum neural network to simultaneously represent the solution and its associated variables. A series of numerical experiments demonstrates that the proposed QPINN framework accurately captures the behavior of nonlinear IDEs and FIPDEs and outperforms classical physics-informed neural networks.
Quantum physics-informed neural networks (QPINNs) solve partial differential equations (PDEs) by training parameterized quantum circuits against physics-based residuals, yet the role of the embedding that maps coordinates into quantum states remains insufficiently understood. In this research, we introduce a unified embedding framework that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and we propose two embeddings within it. The Linear Quantum Neural Network Trainable Embedding QPINN (LQNN-TE-QPINN) generates data-dependent features with an auxiliary quantum circuit and adds them to the input coordinates, instead of multiplying them as in our previous formulation. The adaptive-frequency QPINN (AdaFreq-QPINN) promotes the frequency-scaling factors of an arccos-Chebyshev encoding to trainable parameters. We evaluate both against direct and fixed analytical frequency encodings and alternative hybrid architectures on one- and two-dimensional Burgers equations. In one dimension, LQNN-TE-QPINN achieves a relative L2L_2L2​ error of 0.0916 with 720 trainable parameters, compared with 0.4501 for direct-encoding QPINN and 0.1233 for a locally adaptive classical physics-informed neural network with approximately eleven times more parameters, while AdaFreq-QPINN attains the lowest error among the analytical embeddings with only four additional parameters. In two dimensions, LQNN-TE-QPINN attains the lowest training objective, although its solution error is not clearly separated from that of a data re-uploading QPINN. Under simulated hardware noise, LQNN-TE-QPINN retains the lowest absolute derivative error while degrading most relative to its noiseless baseline. These results demonstrate that embedding design substantially influences the approximation and optimization behavior of QPINNs.
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.