ICON₁₀: Symbolic Discovery Engine for Constitutive Laws

1. The Inverse Problem in Constitutive Modelling. Classical continuum mechanics assumes known constitutive relations (e.g., Newtonian viscosity). However, nanoscale confinement induces emergent transport phenomena—non-local stress coupling, viscosity stratification, and anomalous diffusion—that phenomenological closures cannot capture. The central challenge is the structural inverse problem: given noisy stress-strain observations from Molecular Dynamics, identify the functional form of the constitutive operator without imposing a priori assumptions. 2. Sparse Identification Framework This engine implements a data-driven discovery pipeline based on sparse optimisation. - Sequential Thresholded Ridge Regression (STRidge): Solves the sparse regression problem by combining L₂ regularisation with iterative hard thresholding, avoiding the coefficient shrinkage of LASSO whilst ensuring accurate parameter recovery.- Overcomplete Library Construction: Candidate operators include local terms (S, S², S³, ∇S, τ₀) and non-local integral kernels with varying correlation lengths ξ ∈ {0.02, 0.05, 0.1, 0.2, 0.5}.- Bootstrap Uncertainty Quantification: 500 resamples provide 95% confidence intervals with coverage validation, ensuring frequentist guarantees on discovered parameters. 3. Validation Results (Two Blind Closure Tests). A. Local Constitutive Law Recovery. Synthetic Newtonian fluid data (τ = 2μS) with 5% Gaussian noise. The algorithm converges in 2 iterations to the unique physical form, rejecting all spurious terms. Discovered viscosity: μ = 2.151 vs ground truth μ = 2.150 (relative error 0.04%). Bootstrap 95% CI: [2.13, 2.17] with confirmed coverage. R² = 0.9943; sparsity 80%. B. Non-local Kernel Detection. Synthetic integral data τ(y) = ∫K_ξ(y-y')S(y')dy' with exponential kernel K(r) = ξ⁻¹exp(-|r|/ξ). The algorithm identifies the correct correlation length ξ = 0.1 with 0% error, whilst rejecting all local terms (S, S², S³). Kernel L² error < 10⁻⁶; R² = 0.9981. 4. Theoretical Implications. The successful discrimination between local and non-local constitutive physics from stochastic observations validates the neuro-symbolic approach. The framework demonstrates that sparse regression can automatically determine whether momentum transport follows a pointwise differential operator or a spatially extended integral operator. This capability is essential for discovering emergent constitutive laws in nanoconfined systems where the governing physics is unknown a priori.