pinns-mpf-workflow

PINNs-MPF Workflow

Description

This skill covers the PINNs-MPF approach: coordinate-based physics-informed neural networks that solve multiphase-field (MPF) interface dynamics. The skill includes:

• the pairwise MPF equation and its nondimensionalization;
• extended domain decomposition over space, time, and phases;
• the master/worker network organization, pyramidal training, and transfer of learning;
• interface-focused adaptive collocation (AMFO);
• the Adam/L-BFGS "wheel of optimizers";
• driven-interface, curvature-shrinkage, and triple-junction benchmarks;
• independent comparison against a conventional finite-difference solver.

Invoke this skill when setting up, reproducing, or auditing PINNs-MPF-style simulations, or when deciding whether a coordinate-network PINN is appropriate for a multiphase interface problem.

Domain Context

PINNs-MPF (Elfetni & Darvishi Kamachali, Engineering Analysis with Boundary Elements 176, 2025, peer reviewed) solves the isotropic pairwise multiphase-field equation with networks mapping (x, y, t) to phase fractions. The published formulation is:

∂φ_α/∂t = (μσ/N) Σ_β [ ∇²φ_α − ∇²φ_β + (π²/2η²)(φ_α − φ_β) ] + (μ/N) Σ_β ΔG(φ_α, φ_β)

• Symbols: η is the interface width, μ the interface mobility, σ the interface energy, and N the number of locally present phases. This is the Steinbach/OpenPhase-type obstacle-potential MPF family.
• Coefficient convention (check this before implementing). The obstacle coefficient is printed as π²/(2η²) in Eq. 1 above. The paper's expanded form uses I_α = ∇²φ_α + (π²/η²)φ_α (Eq. 15), i.e. σ(I_α − I_β), and its two-phase form (Eq. 17) is consistent with Eq. 15. The dimensionless Eqs. 22 and 24 scale differently again.
  • The reference solver in Workflow 6 uses the Eq. 15/17 convention. That convention makes the sine profile of width η stationary, and it reproduced both v = μσκ and v = μΔg in this skill's tests.
  • Its driving term is normalized so the N = 2 limit equals Eq. 17.
  • [EXPERT REVIEW NEEDED: reconcile the obstacle and driving-force coefficients of Eqs. 1, 15, 17, 22 and 24 with the code that produced the published results]
• Nondimensional time scale: T = η²/(μσ).
• Two-phase limit: the driven case reduces to a scalar equation with driving term proportional to π√(φ(1−φ))/η. Its sharp-interface limits are v = μΔg for a flat driven front and v = μσκ for curvature-driven motion.