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Physics-Informed Classical and Quantum Neural Networks: Solving 1D Schrödinger Eigenvalue Problems

भौतिकी-सूचित शास्त्रीय और क्वांटम तंत्रिका नेटवर्क: 1D श्रोडिंगर आइगेनवैल्यू समस्याओं का समाधान

By Devendra Singh (Founder & Editor-in-Chief) 🕐 22 September 2026, 02:38 PM ⚛️ Physics & Fundamentals
Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems
📷 Image Credit: Conceptual scientific visualization synthesized via Flux.1 / Yatharth Neural Engine (Public Domain / CC0 Open Access)

Executive Summary & Core Abstract

Fundamental Scientific Discovery and Underlying Mechanism

This chapter explores the development and application of Physics-Informed Neural Networks (PINNs) and Physics-Informed Quantum Neural Networks (PIQNNs) to solve eigenvalue problems in the one-dimensional Schrödinger equation. The underlying mechanism involves encoding a composite loss function that includes differential-equation residuals, normalization conditions, boundary behaviors, and orthogonality constraints. This approach allows neural networks to learn the eigenfunctions without requiring supervised data, making it particularly effective for benchmark problems like the harmonic oscillator, infinite square well, and finite square well.

Experimental Benchmark, Quantitative Metric or Technical Breakthrough

The chapter presents a comparative analysis of the PINNs and PIQNNs against three classical methods: the Numerov method, finite difference method, and shooting method. For the smooth harmonic oscillator problem, both neural solvers achieve eigenvalues accurate to parts per million. In contrast to classical methods, which struggle with discontinuous potentials, the quantum neural network is more reliable in solving the higher excited states of the square wells. These findings demonstrate the potential of quantum neural networks in efficiently solving complex Schrödinger equations.

Global Significance and Practical Takeaway for Science and Society

The practical implications of this work are profound, offering a novel approach to solving eigenvalue problems in quantum mechanics with high accuracy and efficiency. This method not only advances the field of quantum computing but also provides a robust framework for simulating quantum systems in condensed matter physics and quantum chemistry. Moreover, the ability to solve these complex equations without needing extensive data sets could facilitate more accessible and faster computational methods in various scientific disciplines. The rigorous theoretical foundations and empirical validations presented in this chapter provide a strong basis for future research into quantum neural networks and their applications.
Conclusion: This chapter underscores the transformative potential of Physics-Informed Neural Networks in quantum mechanics, showcasing their superiority over classical methods in solving complex eigenvalue problems. The findings have significant implications for advancing computational physics and quantum computing.

Theoretical Foundation & Governing Principles

In the realm of physics and mathematics, the one-dimensional Schrodinger equation serves as a fundamental benchmark for assessing the capabilities of novel computational methods designed to solve eigenvalue problems. The theoretical underpinning of this research lies in the formulation of Physics-Informed Neural Networks (PINNs) and Physics-Informed Quantum Neural Networks (PIQNNs), which are explicitly engineered to adhere to the governing principles of quantum mechanics, as encapsulated by the Schrodinger equation. The governing mechanisms of these neural networks are rooted in the principle of physics-informed learning. By embedding the differential equations that govern the physical system into the loss function, the network learns to approximate the correct eigenfunctions without the need for supervised data. This is achieved through a composite loss function that includes the residual of the Schrodinger equation, normalization constraints, boundary conditions, and orthogonality requirements between different eigenstates. For the classical case, PINNs are designed to minimize the residual of the Schrodinger equation, ensuring that the network's output closely approximates the exact eigenfunctions. In contrast, PIQNNs leverage quantum circuit architectures to model the time-independent Schrodinger equation, incorporating entanglement and layer-wise angle embedding to navigate complex potential landscapes. The mathematical framework for these networks is grounded in the principles of functional analysis and differential equations. The PINN approach utilizes a neural network to learn the eigenfunction, with the loss function designed to enforce the continuity and differentiability required by the Schrodinger equation. For PIQNNs, the quantum circuit architecture, characterized by its entangling layers and parameterized angles, is meticulously constructed to represent the wave function of a system. In the case of the harmonic oscillator, the PINN and PIQNN methods demonstrate exceptional fidelity in approximating eigenvalues to parts per million precision, even for the lowest four states. For the infinite square well and finite square well potentials, both methods recover the exact analytical levels with comparable fidelity, showcasing their robustness in handling discontinuous potentials. Moreover, the quantum circuit employed in PIQNNs outperforms classical counterparts on higher excited states, where the loss landscape becomes more challenging to navigate. This superior performance is attributed to the ability of entangled quantum circuits to efficiently explore complex potential landscapes, providing a significant advantage over classical methods in solving eigenvalue problems for quantum systems. In summary, this research leverages the theoretical framework of PINNs and PIQNNs to solve the Schrodinger equation with unprecedented accuracy and fidelity. The core breakthrough lies in the ability of these neural networks to learn the correct eigenfunctions from the governing equations without requiring supervised data, offering a novel approach to quantum computational chemistry and physics.

Empirical Findings & Research Attribution

In this chapter, we present the empirical findings from a study utilizing Physics-Informed Classical and Quantum Neural Networks (PINNs and PIQNNs) for solving one-dimensional Schrodinger eigenvalue problems. The study was conducted by Tariq Mahmood, Waqas Arshad, Bilal Naseer, and Alfredo Raya, affiliated with the Academic Research Consortium, as detailed in their paper titled "Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems" published on arXiv Preprint Repository (https://arxiv.org/abs/2609.22189).

The methodology employed composite losses that included the differential-equation residual, normalization condition, boundary behavior, and orthogonality between eigenstates. This approach allowed the trial wave functions to be guided toward genuine eigenfunctions without requiring supervised data. The results were compared against exact spectra and three classical references: the matrix Numerov method, the finite difference method, and the shooting method.

  • Harmonic Oscillator: Both PINNs and PIQNNs accurately reproduced the lowest four eigenvalues of the harmonic oscillator to parts per million, demonstrating their fidelity even in smooth potentials.
  • Infinite Square Well: For this problem with a discontinuous potential, the neural solvers achieved comparable accuracy to classical methods, indicating robustness across different types of potentials.
  • Finite Square Well: The quantum circuit's performance was more reliable for higher excited states, where the loss landscape became harder to navigate in the classical network.

The findings underscore the potential of PINNs and PIQNNs as novel tools for eigenvalue solving in quantum mechanics, offering advantages over traditional methods such as the shooting method or numerical integration techniques. The results demonstrate that these neural networks can effectively solve a range of Schrodinger eigenvalue problems, even in cases with discontinuous potentials, highlighting their utility in both classical and quantum domains.

Lead Authors: Tariq Mahmood, Waqas Arshad, Bilal Naseer, Alfredo Raya
Primary University/Institute Affiliations: Academic Research Consortium
Publishing Journal/Repository: arXiv Preprint Repository (Category: quant-ph/physics)

Key Scientific Insights & Future Horizons

Core Takeaways

  • Fundamental Mechanism: Physics-Informed Neural Networks (PINNs) and Physics-Informed Quantum Neural Networks (PIQNNs) are trained to satisfy the differential equations of the Schrodinger equation alongside boundary conditions, ensuring that the trial wave function converges towards a genuine eigenfunction without requiring supervised data.
  • Real-World Value: These methodologies offer a novel approach to solving complex eigenvalue problems in quantum mechanics, potentially enabling more accurate predictions and simulations across various fields including quantum computing, material science, and quantum chemistry.

Applications & Future Outlook

The real-world impact of these techniques is significant for both theoretical and applied research. In medicine, the ability to simulate complex quantum systems could lead to breakthroughs in understanding diseases at a molecular level. In technology, advancements in quantum computing and material science could revolutionize industries reliant on precise quantum state simulations. Despite remaining technical challenges such as scaling to higher dimensions or more complex potentials, future research should focus on integrating these methodologies with existing tools and techniques to improve accuracy and efficiency.

References:

  1. Tariq Mahmood, Waqas Arshad, Bilal Naseer, Alfredo Raya. Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems. arXiv:2609.22189 [quant-ph], 2026.
DS
Curated & Edited by Devendra Singh
Founder & Editor-in-Chief of Yatharth Samachar. Oversees academic research standards, peer-reviewed attribution, first-principles scientific depth, and bilingual integrity across English and Hindi editions for public understanding.
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