Deep learning-based prediction of time-resolved adhesive forces in viscoelastic Hertzian contacts

Technical Summary: Deep Learning-Based Prediction of Time-Resolved Adhesive Forces in Viscoelastic Hertzian Contacts

1. Problem Statement

The accurate prediction of time-resolved adhesive force trajectories in soft viscoelastic contacts is a critical bottleneck for applications in soft robotics, gripping, and manipulation. While the complete force history (including loading, dwell, and unloading phases) encodes essential information regarding energy dissipation, contact stability, and snap-off timing, obtaining this data traditionally requires full numerical simulations using the Boundary Element Method (BEM).

These simulations are computationally expensive, with costs scaling non-linearly with problem complexity (e.g., Tabor parameter, indentation depth, and unloading rate). A single simulation can take seconds to hours, making real-time control loops, iterative design optimization, and inverse identification impractical. Furthermore, analytical models (such as JKR or DMT) are limited to rate-independent elastic limits or specific quasi-static regimes and cannot provide the full force trajectory for arbitrary loading histories involving viscoelastic memory.

2. Methodology

The authors propose a data-driven surrogate model to predict the full adhesive force trajectory from a prescribed displacement history and the Tabor parameter ($\mu$).

Data Generation and Representation:

• Dataset: A dataset of 12,450 trajectories was generated using BEM simulations combined with a Newton-Raphson time-marching scheme. The data spans four orders of magnitude in loading/unloading rates, varied dwell times, and Tabor parameters ranging from 0.2 (DMT-like) to 3.2 (JKR-like).
• Fixed Measurement Step (FMS) Representation: To address the challenge of heterogeneous sequence lengths (where different simulations terminate at different time steps), the authors introduced an FMS representation. Variable-length BEM trajectories are resampled via spline interpolation onto a fixed grid of $N=120$ points (50 for loading, 20 for dwell, 50 for unloading). This preserves physical time information while enabling batch learning.
• Input Features: The model utilizes a physics-guided input representation consisting of four channels: normalized time ($\hat{t}$), normalized indentation ($\hat{\delta}$), causal velocity ($\dot{\hat{\delta}}$), and an edge-detection vector ($e_j$) marking transitions between loading, dwell, and unloading phases. The Tabor parameter ($\mu$) is provided as a static scalar.

Model Architecture:

• Sequence-to-Sequence Modeling: Recognizing that viscoelastic adhesion is governed by a Boltzmann convolution integral (history-dependent), the study employs stateful deep sequence models.
• Architecture Search: Six families of neural networks were evaluated:
  • M1: LSTM (Long Short-Term Memory)
  • M2: LSTM with layer normalization
  • M3: 1D Convolution followed by LSTM
  • M4: Pure Temporal Convolutional Network (TCN)
  • M5: Transformer Encoder
  • M6: Residual LSTM
• Conditioning Mechanisms: Each architecture was tested with three methods of incorporating the static Tabor parameter: concatenation, FiLM (Feature-wise Linear Modulation), and gating.
• Training: Models were trained using the Adam optimizer with Mean Squared Error (MSE) loss on transformed and standardized data. The best-performing configuration was identified through a comparison of global waveform metrics (MSE, MAE, DTW-RMSE) and parameter count.

3. Key Results

• Best Performing Model: The M1-concat architecture (a two-layer LSTM with concatenated Tabor conditioning and a TimeDistributed Dense output layer) achieved the best balance between accuracy and complexity. It attained a held-out MSE of $5.0 \times 10^{-4}$ and a DTW-RMSE of $1.48 \times 10^{-2}$.
• Prediction Accuracy:
  • Pull-off Force: Median error of approximately 2.2%.
  • Hysteresis Area: Median error of approximately 1.1%.
  • Pull-off Time: Median error below 1%.
  • Even in the highest-error quartile (Q4), the mean pull-off error remained at 6.1%, indicating that physically central quantities are robustly captured even when waveform reconstruction has minor deviations.
• Computational Efficiency: The surrogate model provides inference times of approximately 0.16–0.26 seconds for a full force trajectory, regardless of the input parameters. This represents a speed-up of roughly three orders of magnitude compared to the BEM simulations, which ranged from seconds to hours depending on the regime.
• Physical Validation: In the fully relaxed, short-range adhesion limit (JKR-like), the model's predictions of rate-dependent effective surface energy ($\Delta\gamma_{eff}$) closely matched both previous BEM results and the Persson-Brener analytical theory, validating its ability to recover rate-dependent physics without explicit constitutive constraints in the network output.

4. Significance and Claims

The paper claims that deep sequence models can effectively replace time-consuming numerical simulations for predicting the complete time-resolved force response of viscoelastic adhesive contacts. The significance of the work is framed around several key points:

• Representation over Complexity: The study demonstrates that architectural simplicity (a standard LSTM) combined with clear representation of the loading history and adhesion regime is more effective than complex architectures (like Transformers) for this specific physics problem.
• Value of Physics-Guided Features: The inclusion of physically meaningful input features (causal velocity and edge detection) significantly reduced validation error (by a factor of ~7) compared to a minimal time-indentation representation, highlighting that feature engineering remains crucial even in data-driven settings.
• Controlled Error for Real-Time Applications: The surrogate provides a "first-of-its-kind" dataset and model capable of rapid evaluation. The authors position this not as a replacement for high-fidelity simulations in final design verification, but as a tool for rapid exploration, control-oriented applications, and inverse design where repeated queries are necessary.
• Limitations: The authors modestly note that the model is trained on a specific single-relaxation viscoelastic model (Standard Linear Solid) and cannot automatically extrapolate to materially different rheologies. Additionally, the highest errors are concentrated in trajectories with sharp transitions and near-instability events, suggesting that future datasets should prioritize these regimes.

In conclusion, the work establishes a viable pathway for integrating contact state estimation into real-time soft robotic systems, moving beyond scalar detachment metrics to full trajectory prediction.