A research-oriented study of physics-informed neural networks for stationary and time-dependent quantum systems, with emphasis on whether governing equations, normalization constraints, and physically motivated regularization are sufficient to recover accurate quantum solutions when dense supervision is limited or unavailable. The project combines a high-accuracy harmonic-oscillator eigenvalue study, a complex-valued time-dependent Schrödinger propagation study, and a comparative multi-problem benchmark that evaluates transferability, architecture sensitivity, and robustness under a shared experimental framework.
Relative L² error for the QHO ground state, with learned energy 0.50001526 and absolute energy error 1.52588 × 10−5.
Initial density relative L² error for Gaussian wavepacket propagation, with mean density error 4.18% across the full spacetime interval.
Best shared benchmark architecture in the combined notebook, reaching relative L² error 0.26585 with stable noise behavior from 0.25654 to 0.25035.
This project presents a research-oriented framework for applying Physics-Informed Neural Networks (PINNs) to quantum-relevant physical systems. The central objective is to determine whether governing equations, normalization structure, and physically motivated consistency terms can act as a sufficient supervisory signal for recovering high-quality quantum solutions in regimes where dense labeled data is unnecessary, scarce, or expensive to generate.
The empirical argument is organized around three complementary notebooks. The harmonic-oscillator notebook establishes the strongest stationary-state result, the time-dependent Schrödinger notebook establishes the strongest propagation result, and the combined benchmark notebook provides the broadest methodological evidence by quantifying transferability, architecture sensitivity, and robustness across multiple quantum regimes.
The project's strongest stationary-state result is the QHO ground-state benchmark: relative L² error 1.56934 × 10^-3, max pointwise error 1.25322 × 10^-3, learned energy 0.50001526, absolute energy error 1.52588 × 10^-5, squared overlap 0.99999754, and Rayleigh-consistency gap 3.59893 × 10^-6.
The strongest time-dependent result is the Gaussian wavepacket benchmark: initial density relative L² error 8.0 × 10^-8, mean density relative L² error 4.18%, final-time density relative L² error 6.71%, norm range [0.990997, 1.010868], and mean Ehrenfest errors 2.56 × 10^-2 and 2.99 × 10^-2.
The project's broadest contribution is the integrative benchmark layer: a unified four-problem study with architecture, scaling, and noise ablations. In the executed sweep, the best shared configuration is 5 layers × 64 units with relative L² error 0.26585, while the noise study varies by only 0.00619 across amplitudes 0.00 to 0.20.
The root website now highlights the strongest results from the three notebook studies through representative figures drawn directly from the executed benchmark outputs.
This figure summarizes the strongest stationary-state result in the project: relative L² error 1.56934 × 10^-3, max pointwise error 1.25322 × 10^-3, learned energy 0.50001526, and absolute energy error 1.52588 × 10^-5.
This density map highlights the strongest time-dependent result in the project, where the benchmark reaches 8.0 × 10^-8 initial density rel-L² error, 4.18% mean density rel-L² error, and 6.71% final-time rel-L² error.
This figure complements the TDSE density benchmark by showing the physical diagnostics that support the result: norm range [0.990997, 1.010868] together with mean Ehrenfest errors 2.56 × 10^-2 and 2.99 × 10^-2.
This figure highlights the main integrative contribution of the combined notebook: under a shared benchmark protocol, the strongest architecture in the executed sweep is 5 layers × 64 units with relative L² error 0.26585.
Configure a representative quantum problem, vary core training parameters, and inspect a lightweight simulated view of how PINN metrics respond to architectural and optimization choices.
Fully executed HTML notebooks with figures, equations, benchmark tables, and interpretive narrative. These documents form the primary evidence layer of the project and should be read as complementary studies rather than interchangeable tutorials.
Direct simulation of quantum systems is computationally expensive, and experimental data from quantum platforms is frequently sparse, noisy, or constrained by hardware limitations. These conditions motivate a learning-theoretic approach in which physical structure is not added after training, but built directly into the optimization problem as an inductive bias.
PDE residuals are enforced as soft constraints through residual terms in the training objective, turning the governing equation itself into the dominant source of supervision.
Exact spatial and temporal derivatives are computed via reverse-mode autodiff, eliminating finite-difference approximation error.
Collocation points augment limited measurements, reducing dependence on dense observational sampling while preserving agreement with known physical structure.
GPU-compatible training supports larger collocation grids and broader architecture sweeps without changing the underlying scientific workflow.
Each target system is specified by a governing PDE or ODE, boundary and initial conditions, domain geometry, and optional measurement data. The formulation determines the structure of the physics residual and the data-consistency term in the composite loss.
A deep neural network approximates the solution field — e.g., wavefunction ψ(x, t) or probability density |ψ|². Architectures are then specialized when needed: the project's strongest stationary-state result comes from an accuracy-oriented harmonic-oscillator configuration, while the strongest time-dependent result comes from a dual-output ComplexPINN with hard initial conditioning and analytic anchors.
The PDE residual is evaluated at a set of interior collocation points sampled from the domain, without requiring a mesh. Boundary and initial condition points are sampled separately and weighted in the composite loss.
The total loss combines the PDE residual, boundary condition violation, initial condition error, and optional data mismatch terms with tunable weights. Optimization proceeds via Adam or L-BFGS with exact gradient computation through autograd.
Metrics tracked include PDE residual norm, relative L² error against analytic or numerical reference, energy consistency, norm preservation, Ehrenfest agreement, convergence under varying collocation density, and stability under observation noise.
Solve the time-dependent Schrödinger equation for a particle in a potential well. In the executed benchmark, the model learns Gaussian wavepacket evolution with 8.0 × 10^-8 initial density rel-L² error and controlled transport diagnostics across the full time window.
Approximate energy eigenstates and eigenvalues under orthonormality constraints, recovering the QHO ground state with rel-L² 1.56934 × 10^-3 and an absolute energy error of 1.52588 × 10^-5.
Extend to quartic and double-well potentials to study the impact of nonlinearity on PINN convergence and solution accuracy, and to place those systems inside a unified comparative benchmark with shared architecture and noise studies.
Learn trajectory solutions under energy conservation constraints, serving as a controlled benchmark for comparing physical agreement across collocation densities and network architectures.
# 1. Install dependencies pip install -r requirements.txt # 2. Train a PINN on the harmonic oscillator python -m src.train \ --problem harmonic_oscillator \ --epochs 5000 \ --collocation 2000 \ --model-path model.pt # 3. Start the inference API python -m src.server # 4. Serve the demo (separate terminal) python -m http.server 8000 # → open http://localhost:8000
Governing differential equations are penalized at collocation points without a discretization mesh, enabling flexible domain sampling and adaptive refinement.
All spatial and temporal derivatives are computed via PyTorch autograd, eliminating finite-difference approximation error with seamless gradient flow.
Controlled experiments vary the ratio of collocation points to observed data, enabling direct empirical comparison of physical agreement versus data efficiency.
Systematic variation of observation density, noise level, and domain coverage characterizes PINN stability rigorously across quantum system benchmarks.
Full GPU execution support via PyTorch enables experiments at larger network sizes, finer collocation grids, and higher-dimensional parameter spaces.
A lightweight API endpoint and browser interface provide an accessible front end for inspecting model outputs, parameter choices, and qualitative solution behavior.
Add multi-dimensional Schrödinger equations, Gross-Pitaevskii equations for Bose-Einstein condensates, and time-dependent perturbation problems. Improve convergence diagnostics and residual visualization tools.
Implement dynamic collocation strategies that concentrate sampling in high-residual regions. Introduce hard constraint enforcement for boundary and initial conditions via distance-function multipliers.
Extend the framework toward Lindblad master equations and open system dynamics. Explore integration with quantum hardware data for hybrid experimental-computational validation workflows.