ME_218_PINN

Inverse Elasticity PINN via Direct Hybrid Collocation (DHC)

This repository contains the implementation of a Physics-Informed Neural Network (PINN) framework designed for the inverse identification of spatially-varying Young’s moduli in additively manufactured lattice structures. Go to detailed Math about this project

Overview

Traditional Digital Image Correlation (DIC) data assimilation often relies on Radial Basis Function (RBF) pre-interpolation, which can lead to non-physical displacement gradients (e.g., strains exceeding 26,000%) due to a lack of mechanical constraints.

The Direct Hybrid Collocation (DHC) paradigm eliminates RBF pre-processing by employing the neural network itself as the direct spatial interpolator. Sparse DIC blocks act as Dirichlet anchors, while a dense Sobol quasi-random collocation cloud enforces the 2D Cauchy Momentum PDE across the domain.

Key Features

• Bifurcated Architecture: Independent branches for kinematics (space-time) and constitutive behavior (space-only).
• Adaptive Lagrangian Optimization: Training is posed as a constrained saddle-point problem using simultaneous Adam primal descent and dual ascent over six learned Lagrange multipliers.
• Spectral Bias Mitigation: Utilizes Random Fourier Feature (RFF) encoding with distinct scales to capture fine kinematic details while maintaining a smooth modulus field.
• Positivity Enforcement: A softplus offset construction ensures the identified Young’s modulus $E(x,y)$ remains physically valid ($E > 0$).
• Automated Pipeline: Includes Savitzky-Golay temporal smoothing, rolling $R^2$ Hookean regime isolation, and dimensional restoration via the chain rule.

Technical Specifications

| Parameter | Value | Description | | :— | :— | :— | | Epochs | 10,000 | Total training iterations | | Collocation Points | 10,000 | Sobol quasi-random samples | | Optimizer | Adam | Primal/Dual learning rate of $10^{-3}$ | | Activation | SiLU | Smooth $C^{\infty}$ function for higher-order derivatives | | Physics | 2D Plane Stress | Cauchy Momentum Balance |

Installation and Execution (GPU Only)

1. Initialize and Activate Virtual Environment:

   python -m venv venv
   # Windows:
   .\venv\Scripts\activate
   # Linux/macOS:
   source venv/bin/activate
   

2. Install Requirements:

   pip install torch --index-url [https://download.pytorch.org/whl/cpu](https://download.pytorch.org/whl/cpu)
   pip install pandas numpy scipy matplotlib
   

3. Run the Program:

   python A_DHC.py
   

Results

The framework was validated on 80% Lines, 60% Gyroid, and 80% Gyroid specimens.

• Scale Effect Identification: PINN-recovered local stiffness ($E_{PINN}$) was found to be 1.4x to 1.9x higher than macroscopic UTM-averaged compliance ($E_{UTM}$), capturing intermediate-scale deformation physics.
• Stability: Successfully suppressed oscillatory artifacts and non-physical divergences inherent in standard interpolation techniques.

Reference

Van, V., et al. (2026). Direct Hybrid Collocation PINN (DHC-PINN/IE-PINN) for Inverse Identification of Spatially-Varying Elastic Moduli in Additively Manufactured Lattice Structures.

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