Quantum Physics-Informed Neural Networks

• Quantum Physics-Informed Neural Networks are models that integrate neural networks with quantum mechanical laws, enforcing boundary conditions and symmetry via analytical parametrization.
• They employ a composite loss function that combines differential equation residuals with normalization and orthogonality constraints to produce physically admissible, analytical solutions.
• These networks effectively solve quantum eigenvalue problems, such as finite square wells and the hydrogen atom, with potential scalability to complex, high-dimensional systems.

Quantum Physics-Informed Neural Networks (QPINNs) represent a class of models that merge the expressivity of artificial neural networks with explicit encoding of quantum mechanical laws. Originally developed for quantum eigenvalue problems, QPINNs are designed to discover both eigenfunctions and eigenvalues simultaneously in an unsupervised, data-free fashion. The network’s optimization objective enforces satisfaction of quantum differential equations, normalization, and orthogonality constraints, while leveraging symmetry properties and analytically satisfying boundary conditions. This approach facilitates efficient solution discovery for quantum systems such as finite and multiple square wells and the hydrogen atom, and offers a principled extension pathway to more complex, high-dimensional quantum problems.

1. Mathematical and Architectural Foundations

The core QPINN method seeks solutions to eigenvalue problems of the form $\mathcal L f(x) = \lambda f(x)$, e.g., the time-independent Schrödinger equation, where $\mathcal L$ is a Hermitian differential operator. The framework utilizes a feed-forward network that takes the spatial coordinate $x$ (and a trainable constant input $\lambda$ for the eigenvalue) and produces as output an unconstrained function $N(x, \lambda)$: $f(x, \lambda) = f_b + g(x) N(x, \lambda)$ Here, $f_b$ denotes an admissible fixed boundary value, and $g(x)$ is an explicit function such that $f(x, \lambda)$ automatically satisfies Dirichlet or other specified boundary conditions (e.g., $g(x) = (1-e^{-(x-x_L)})(1-e^{-(x-x_R)})$ for vanishing endpoints). This parametric trick yields analytical, differentiable solutions and circumvents numerical errors associated with conventional PDE discretization.

Network symmetry is embedded by processing $x$ and $-x$ through parallel paths and combining their outputs (sum for even, difference for odd solution parity), reducing the effective function space and accelerating convergence. The “eigenvalue” $\lambda$ itself is treated as a network parameter, updated by backpropagation.

2. Physics-Informed Loss Construction

The QPINN framework enforces quantum physics constraints via a composite loss function: $L = L_{DE} + L_{reg}$ where

$$L_{DE} = \frac{1}{M} \sum_{i=1}^M [\mathcal L f(x_i, \lambda) - \lambda f(x_i, \lambda)]^2$$

is the main physics-informed PDE residual. $L_{reg}$ incorporates additional regularization explicitly grounded in quantum requirements:

• Normalization (norm-loss): Enforces $\|f\|^2 = 1$ (or other target norm) by penalizing deviations, i.e.,

$$L_\text{norm} = \Big( \langle f, f \rangle - \frac{M}{x_R - x_L} \Big)^2$$

This term prevents trivial solutions ($f \equiv 0$) and aligns with quantum probability conservation.

• Orthogonality (ortho-loss): Drives newly found eigenfunctions toward orthogonality with previously discovered ones. For the $k$-th eigenfunction $\psi_k$ and previously obtained collection $\{\psi_j\}_{j<k}$,

$$L_\text{orth} = \sum_{j<k} \langle \psi_k, \psi_j \rangle^2$$

Upon satisfaction of a patience criterion, the new solution is added to the running orthogonality constraint for future optimization.

These loss terms replace ad hoc penalization schemes and naturally encode Hermitian eigensystem properties.

3. Embedding Quantum Symmetries in the Network

Symmetry in quantum mechanics often implies that permissible eigenfunctions have definite parity. To incorporate this prior, the QPINN architecture processes $x$ and $-x$ via two identical subnetworks and then sums (enforcing even symmetry) or subtracts (enforcing odd symmetry) their outputs. This reduces the hypothesis space and yields

• Improved convergence rates,
• Higher final accuracy,
• Physical admissibility (only even/odd eigenfunctions are considered for problems with such symmetry).

Comparative results in the studied quantum wells and hydrogen atom systems demonstrate that symmetry-embedded networks converge substantially faster and more reliably than those lacking explicit symmetry.

4. Application to Quantum Eigenvalue Problems

QPINNs have been used to solve several canonical quantum mechanics problems:

• Finite Square Well: The network finds lowest-energy eigenfunctions and eigenvalues for $V(x) = 0$ within $[0,\ell]$ and $V(x) = V_0$ outside, with parametric $g(x)$ enforcing vanishing at $x=0,\ell$. QPINN-reported energies precisely match analytical results.
• Multiple Square Wells: For potentials comprised of repeated finite wells ($V(x)$ piecewise constant), QPINNs discover orthogonal eigenfunctions for the multi-well structure, demonstrating scalability and flexibility.
• Hydrogen Atom (spherically symmetric radial equation): QPINNs operate on the radial ODE,

$$\frac{d^2R}{dr^2} + \frac{2}{r}\frac{dR}{dr} = -\Big[ \frac{2\mu}{\hbar^2}\left(E + \frac{Ze^2}{4\pi\varepsilon_0 r}\right) - \frac{l(l+1)}{r^2}\Big] R$$

with custom $g(x)$ to enforce boundary behavior at $r=0, \infty$. The method reproduces known eigenvalues and eigenfunctions for discrete quantum numbers $(n,l)$.

The architecture and loss construction are applicable to both single- and multi-well/atom problems and integrate seamlessly with boundary and symmetry constraints.

5. Advantages, Limitations, and Scaling Properties

Advantages:

• Analytical, differentiable solution forms automatically obey boundary conditions.
• Intrinsic enforcement of normalization and orthogonality, robustly avoiding trivial or duplicate solutions.
• No numerical mesh is required; optimization is data-free except for sampled $x_i$, and only predictions from the neural network are used.
• Symmetry embedding sharpens convergence and physical reliability.
• The direct parametrization of $\lambda$ as a network variable removes the need for laborious expectation value evaluations over the domain at each step.

Limitations:

• Careful tuning of regularization coefficients (e.g., for norm-loss, ortho-loss), and patience thresholds is essential for efficient discovery of sequential eigenfunctions.
• Special care must be taken in singular regions (e.g., near $r\to0$ for the hydrogen atom), often necessitating domain truncation to avoid numerical instability.
• The algorithm is designed to find one eigenpair at a time, requiring iteration for full spectrum extraction. While the ortho-loss mitigates repetition, manual management of previously found solutions and sequential minimization steps are still required.

Scaling to higher dimensions and more complex boundary conditions is technically feasible due to the mesh-free, differentiable network properties, although computational resource requirements increase accordingly.

6. Extensions and Future Research Directions

The QPINN strategy as established in this work is not limited to conventional quantum mechanics. The authors suggest the following extensions:

• Higher-dimensional eigensystems: Application to full three-dimensional Schrödinger or Helmholtz equations relevant in atomic/molecular physics and optics, where curse-of-dimensionality issues in meshing and integration are acute.
• Time-dependent Problems: Inclusion of extra network inputs to handle evolution under, e.g., the time-dependent Schrödinger equation, enabling quantum dynamic simulations in an unsupervised setting.
• Sturm–Liouville and Generalized Eigenproblems: Applicability to broader classes of self-adjoint differential equations, including those appearing in electromagnetism and wave propagation.
• Solution Recognition Automation: Improved convergence via dynamic patience conditions and symmetry-switching protocols to detect new eigenfunction discovery, aiming for robust, automated, and efficient spectrum computation.

7. Summary Table

Property	Classical Solver	QPINN Framework	Benefit in QPINN
Boundary Conformance	Weak (penalized)	Identical (parametric)	No boundary violation; mesh-free solution
Normalization	Implicit/ad hoc	Explicit (norm-loss)	No trivial/zero solutions
Orthogonality	Manual/post-process	Explicit (ortho-loss)	Sequential, robust eigenstate discovery
Symmetry Handling	Manual	Embedded in architecture	Accelerated, physically correct convergence
Energy as Network Variable	No (computed each step)	Yes	Fast eigenvalue convergence
Data Requirements	None (integration grid)	None (collocation points only)	Efficient, scalable

Conclusion

Quantum Physics-Informed Neural Networks as formalized in the cited work offer an unsupervised, symmetry-aware methodology for solving quantum eigenvalue problems, yielding analytical solutions that encode both boundary and physical constraints by design. The framework's modularity, flexibility, and direct encoding of quantum principles position it as a robust alternative to traditional numerical methods, and it is readily extensible to more complex quantum, classical, and hybrid eigensystem problems (Jin et al., 2022).