Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems

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.

We present PINNACLE, an open-source computational framework for physics-informed neural networks (PINNs) that integrates modern training strategies, multi-GPU acceleration, and hybrid quantum-classical architectures within a unified modular workflow. The framework enables systematic evaluation of PINN performance across benchmark problems including 1D hyperbolic conservation laws, incompressible flows, and electromagnetic wave propagation. It supports a range of architectural and training enhancements, including Fourier feature embeddings, random weight factorization, strict boundary condition enforcement, adaptive loss balancing, curriculum training, and second-order optimization strategies, with extensibility to additional methods. We provide a comprehensive benchmark study quantifying the impact of these methods on convergence, accuracy, and computational cost, and analyze distributed data parallel scaling in terms of runtime and memory efficiency. In addition, we extend the framework to hybrid quantum-classical PINNs and derive a formal estimate for circuit-evaluation complexity under parameter-shift differentiation. Results highlight the sensitivity of PINNs to architectural and training choices, confirm their high computational cost relative to classical solvers, and identify regimes where hybrid quantum models offer improved parameter efficiency. PINNACLE provides a foundation for benchmarking physics-informed learning methods and guiding future developments through quantitative assessment of their trade-offs.

Full waveform inversion (FWI) reconstructs heterogeneous material properties from receiver data but remains computationally demanding. Physics-informed neural networks (PINNs) and their domain-decomposed variants (FBPINNs) offer a mesh-free alternative but face convergence challenges when representing complex velocity fields. We present a hybrid quantum-classical FBPINN for acoustic FWI, bringing together quantum computing and classical machine learning, in which the decomposed wavefield network and the global velocity network are implemented as classical-to-quantum pipelines terminating in parameterized quantum circuits (PQCs). The PQCs are realized as differentiable JAX statevector simulators, enabling end-to-end automatic differentiation through the classical PINN, the quantum circuit, and the physics-informed loss. On a geophysical anomaly benchmark, the quantum hybrid reaches a lower L1 velocity error than the primary classical FBPINN baseline in approximately 8x fewer training iterations, despite using approximately 33% fewer trainable parameters, and it outperforms all 15 classical hyperparameter variants tested. A second benchmark (checkerboard) demonstrates the generality of the inversion pipeline, confirming that the quantum hybrid architecture can recover structured spatial variations beyond the localized anomaly benchmark. Our framework is broadly applicable to wave-based inverse problems beyond geophysics, including medical ultrasound tomography and non-destructive evaluation.