🧪 Digital Rheology of Polymers Using Physics-Informed LSTM

Manuscript in Preparation (MDPI) | Amrita Vishwa Vidyapeetham, Coimbatore

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📄 Abstract

Viscoelastic materials subjected to Large Amplitude Oscillatory Shear (LAOS) exhibit strong nonlinear stress–strain responses, making data-driven prediction challenging. We develop a hybrid Physics-Informed LSTM (PI-LSTM) model that integrates a two-layer recurrent neural architecture with a Burgers viscoelastic constitutive law.

Training is performed using an Augmented Lagrangian (ADMM-like) framework that enforces physical residuals through adaptive multipliers. The model achieves R² = 0.9819, accurately captures hysteresis loops and peak stresses, and consistently outperforms baseline purely data-driven models.

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🏗️ System Architecture

Input: Strain ε(t), Strain Rate ε̇(t), Time t
          │
          ▼
  ┌─────────────────────────────────┐
  │   2-Layer Unidirectional LSTM   │
  │   (128 hidden units per layer)  │
  └─────────────────────────────────┘
          │  hidden state h_t ∈ R^128
          ▼
  LayerNorm → FC1(128→128, SiLU, dropout=0.2) → FC2(128→1)
          │
          ▼  σ_NL (learned nonlinear stress)
          +
  σ_Linear = softplus(E₂) · ε   (physics-based linear term)
          │
          ▼
  σ_pred = σ_NL + σ_Linear
          │
          ▼
  Physics Residual: r(t) = dσ_NL/dt + σ_NL/λ_eff − G_eff · dε/dt
          │
          ▼
  Augmented Lagrangian Loss:
  L_AL = L_data + Σ M_i·r_i + (ρ/2)·Σ r_i²
          │
          ▼
  AdamW Optimizer + Gradient Clipping + LR Scheduling

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📁 Repository Structure

Digital-Rheology-PI-LSTM/
├── README.md
├── requirements.txt
├── src/
│   ├── pi_lstm_model.py         # PI-LSTM architecture (Burgers + LSTM)
│   ├── burgers_model.py         # Burgers constitutive equation
│   ├── augmented_lagrangian.py  # AL optimization framework
│   ├── train.py                 # Training loop (70 epochs, early stopping ep.67)
│   ├── evaluate.py              # R², RMSE, REC curve, stress distribution
│   └── dataset.py               # LAOS dataset loader (Choi & Rogers)
├── notebooks/
│   └── PI_LSTM_Rheology.ipynb
├── results/
│   ├── validation_loss.png      # Convergence over 70 epochs
│   ├── stress_strain_lissajous.png
│   ├── residual_distribution.png
│   ├── rec_curve.png
│   └── metrics.json             # R²=0.9819, RMSE
├── images/
│   └── pi_lstm_architecture.png
└── LICENSE

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⚙️ Methods

Dataset

• Source: Choi & Rogers — thixotropic fumed silica suspension (R972, Evonik)
• Medium: Paraffin oil + low-MW polybutene (69:31 wt%)
• Concentration: 2.9 vol% fumed silica
• Rheometer: ARES-G2 (TA Instruments), cone-plate geometry (40mm, 2°), 20°C
• Protocol: LAOS (Large Amplitude Oscillatory Shear)

Neural Architecture

Component	Specification
Input sequence length	64 (stride 32)
LSTM layers	2 unidirectional
LSTM hidden units	128 per layer
FC1	128→128, SiLU activation, dropout=0.2
FC2	128→1 (σ_NL output)
Physics term	σ_Linear = softplus(E₂)·ε

Burgers Constitutive Model

σ + (E₁/η₁ + E₂/η₂ + E₂/η₁)·σ̇ + (E₁E₂/η₁η₂)·σ̈ = η₁ε̇ + (E₂η₁/η₁η₂)·ε̈

Parameters E₁, E₂, η₁, η₂ optimized to match experimental LAOS response.

Loss Function

L_AL = L_data + Σ M_i·r_i + (ρ/2)·Σ r_i²

where:
  L_data = (1/N) Σ w_i(σ̂_i - σ_i)²,   w_i = 1/(1 + σ_i²)
  r(t)   = dσ_NL/dt + σ_NL/λ_eff − G_eff·dε/dt
  M_i updated: M_{k+1} = M_k + ρ·r_i

Training

Parameter	Value
Optimizer	AdamW
Epochs	70 (early stopping @ epoch 67)
Gradient clipping	✅
LR scheduling	✅

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📊 Results

Performance Metrics

Metric	PI-LSTM (Proposed)	Baseline LSTM
R²	0.9819	0.91
RMSE	Lower	Higher
Hysteresis loop capture	✅ Accurate	❌ Partial
Peak stress reproduction	✅	❌
Physics constraint satisfied	✅	❌

Key Findings

• Validation loss converges stably — early stopping triggered at epoch 67
• Residual distribution tightly centered near zero — physics constraints respected
• R² = 0.9819 on max-fidelity LAOS test set
• REC curve shows >90% of predictions within 0.5 Pa absolute error tolerance

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🚀 Getting Started

pip install -r requirements.txt
python src/train.py --epochs 70 --hidden 128 --lr 1e-3
python src/evaluate.py --model_path models/pi_lstm.pth

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📦 requirements.txt

torch==2.0.1
numpy==1.24.3
scipy==1.11.1
pandas==2.0.3
matplotlib==3.7.2
scikit-learn==1.3.0
tqdm==4.65.0

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👥 Authors

Name	Affiliation
Deepak Skandh	Amrita Vishwa Vidyapeetham
Nithin S	Amrita Vishwa Vidyapeetham
Akhillesh Varathan	Amrita Vishwa Vidyapeetham
Kavin	Amrita Vishwa Vidyapeetham
Neelesh Ashok (Supervisor)	neelesh@cb.amrita.edu

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📚 Citation

@article{skandh2026pi_lstm,
  title   = {Digital Rheology of Polymers Using Physics-Informed LSTM Approach},
  author  = {Skandh, Deepak and Nithin and Varathan, Akhillesh and Kavin and Ashok, Neelesh},
  journal = {Journal Not Specified (MDPI)},
  year    = {2026},
  note    = {Manuscript in preparation}
}

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📜 Key References

1. Raissi et al. Physics-Informed Neural Networks. J. Comput. Phys., 2019.
2. Hochreiter & Schmidhuber. Long Short-Term Memory. Neural Comput., 1997.
3. Choi & Rogers. Thixotropic fumed silica LAOS dataset, 2025.
4. Boyd et al. ADMM Distributed Optimization. Found. Trends ML, 2011.

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📍 Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu  |  Manuscript In Preparation (MDPI)