# Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty propagation

arXiv: https://arxiv.org/abs/2105.09980 · HTML: https://mlsanity.com/paper/2105.09980/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 106 references that are in the map.
- First posted on arXiv 20 May 2021.

## Builds on (most cited first, 21 of 21)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [A Theoretically Grounded Application of Dropout in Recurrent Neural Networks](https://mlsanity.com/paper/1512.05287/) (2015)
- [Fast Bayesian Optimization of Machine Learning Hyperparameters on Large Datasets](https://mlsanity.com/paper/1605.07079/) (2016)
- [Hyperparameters and Tuning Strategies for Random Forest](https://mlsanity.com/paper/1804.03515/) (2018)
- [Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity](https://mlsanity.com/paper/2001.04292/) (2020)
- [A fast PC algorithm for high dimensional causal discovery with multi-core PCs](https://mlsanity.com/paper/1502.02454/) (2015)
- [Causal Discovery and Forecasting in Nonstationary Environments with State-Space Models](https://mlsanity.com/paper/1905.10857/) (2019)
- [Meta-modeling game for deriving theoretical-consistent, micro-structural-based traction-separation laws via deep reinforcement learning](https://mlsanity.com/paper/1810.10535/) (2018)
- [A Physics-Constrained Data-Driven Approach Based on Locally Convex Reconstruction for Noisy Database](https://mlsanity.com/paper/1907.12651/) (2019)
- [Learnable Bernoulli Dropout for Bayesian Deep Learning](https://mlsanity.com/paper/2002.05155/) (2020)
- [Improved Dropout for Shallow and Deep Learning](https://mlsanity.com/paper/1602.02220/) (2016)
- [A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with AI-guided experimentation](https://mlsanity.com/paper/1903.04307/) (2019)
- [A non-cooperative meta-modeling game for automated third-party calibrating, validating, and falsifying constitutive laws with parallelized adversarial attacks](https://mlsanity.com/paper/2004.09392/) (2020)
- [An accelerated hybrid data-driven/model-based approach for poroelasticity problems with multi-fidelity multi-physics data](https://mlsanity.com/paper/2012.00165/) (2020)
- [Bayesian Projected Calibration of Computer Models](https://mlsanity.com/paper/1803.01231/) (2018)

## Built on by (most cited first, 10 of 10)

- [Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics](https://mlsanity.com/paper/2304.14369/) (2023)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Denoising diffusion algorithm for inverse design of microstructures with fine-tuned nonlinear material properties](https://mlsanity.com/paper/2302.12881/) (2023)
- [Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)](https://mlsanity.com/paper/2108.13137/) (2021)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Geometric deep learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticity](https://mlsanity.com/paper/2208.00246/) (2022)
- [Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions](https://mlsanity.com/paper/2307.13149/) (2023)
- [Investigating Deep Learning Model Calibration for Classification Problems in Mechanics](https://mlsanity.com/paper/2212.00881/) (2022)
- [Learning 3D Garment Animation from Trajectories of A Piece of Cloth](https://mlsanity.com/paper/2501.01393/) (2025)
- [Towards scientific machine learning for granular material simulations -- challenges and opportunities](https://mlsanity.com/paper/2504.08766/) (2025)

## Abstract (arXiv)

This paper presents a computational framework that generates ensemble predictive mechanics models with uncertainty quantification (UQ). We first develop a causal discovery algorithm to infer causal relations among time-history data measured during each representative volume element (RVE) simulation through a directed acyclic graph (DAG). With multiple plausible sets of causal relationships estimated from multiple RVE simulations, the predictions are propagated in the derived causal graph while using a deep neural network equipped with dropout layers as a Bayesian approximation for uncertainty quantification. We select two representative numerical examples (traction-separation laws for frictional interfaces, elastoplasticity models for granular assembles) to examine the accuracy and robustness of the proposed causal discovery method for the common material law predictions in civil engineering applications.