Demonstration of AI-Assisted Scientific Workflow on Canonical Benchmarks

Abstract

We present a fully reproducible demonstration of an AI-assisted scientific workflow designed for a broad physics, mathematics, and computer-science readership. The initial project artifact stack was generated from one single user prompt and then reviewed and curated for submission by the human author. Rather than claiming a new scientific discovery, the manuscript uses canonical benchmark problems with exact, manufactured, or independently checkable answers. The analytical component starts from the one-dimensional quantum harmonic oscillator, derives its dimensionless form, and validates finite-difference eigenpairs against exact Hermite-function benchmarks. The numerical partial-differential-equation component solves a heat equation with a known modal solution and a Poisson problem verified by a manufactured solution, with explicit convergence studies. The inverse-modeling component fits synthetic damped-oscillation data by nonlinear least squares and quantifies parametric uncertainty by bootstrap resampling. The computational-science component compares dense and sparse eigensolvers and contrasts direct and iterative sparse linear solvers, with careful interpretation of machine-dependent timing data. Taken together, the results show that contemporary AI can already serve as a useful scientific copilot for derivation, implementation, validation, visualization, and manuscript preparation, provided that each stage is constrained by benchmark theory, explicit verification, and transparent artifacts. The demonstration is therefore relevant not because the underlying science is novel, but because it offers a concrete template for trustworthy AI use in technical research practice.

Figures (5)

• Figure 1: End-to-end structure of the demonstration workflow. The boxes denote the main stages of the project: problem definition, analytic derivation, numerical implementation, verification, data and figure generation, manuscript assembly, and reproducible packaging. The arrows indicate that the workflow is not merely sequential; benchmark formulas and numerical checks constrain downstream coding, and failed checks feed back to earlier stages. The figure is deliberately sober rather than infographic-like because the paper's claim concerns workflow discipline, not interface design.
• Figure 2: Validation of the harmonic-oscillator case study. (a) Exact eigenvalues $E_n=n+1/2$ (black solid circles) and finite-difference results on the finest grid (red dashed squares). (b) Exact Hermite-function eigenstates (black solid) and numerical eigenvectors (red dashed), shown with vertical offsets for $n=0,1,2,3$. (c) Eigenvalue errors versus grid spacing for the first four levels. (d) Discrete $L^2$ errors of the corresponding eigenfunctions versus grid spacing. The figure demonstrates that the AI-assisted implementation reproduces a canonical quantum benchmark with the correct second-order convergence.
• Figure 3: PDE validation results. (a) Heat-equation snapshots at several times, with exact solutions shown by black solid lines and Crank--Nicolson results by red dashed lines. (b) Heat-equation terminal $L^2$ errors versus grid spacing for Crank--Nicolson and FTCS. (c) Numerical solution of the manufactured Poisson problem on a representative grid. (d) Poisson $L^2$ and $L^\infty$ errors versus mesh spacing. The figure demonstrates that the generated solvers agree with exact or manufactured benchmarks and exhibit the expected second-order behavior.
• Figure 4: Inverse-modeling demonstration for noisy damped oscillations. (a) Ground-truth signal (black solid), noisy observations with error bars (gray markers), nonlinear least-squares fit (red solid), and bootstrap $95\%$ prediction band (gray shaded region). (b) Residuals of the best fit as a function of time. (c) Fitted parameter values with bootstrap intervals compared with ground truth. The figure shows that the AI-assisted workflow can formulate the inverse problem, run the regression, and present uncertainty information, while still requiring explicit statistical checks.
• Figure 5: Algorithmic comparisons. (a) Runtime versus problem size for dense diagonalization and sparse low-mode eigensolving of the harmonic-oscillator matrix. (b) Runtime versus problem size for a sparse direct solve and conjugate gradients on the manufactured Poisson problem. Both panels should be interpreted as machine-dependent workflow demonstrations rather than universal performance claims. The main conclusion is that AI can help construct and document algorithmic comparisons in a reproducible form.