Introduction

Inverse modeling for granular flows is essential in landslide and debris flow hazard analysis, helping in understanding flow dynamics and estimating critical parameters, which are key for risk assessment, model calibration, and developing effective mitigation strategies. Specifically, in granular flow scenarios, it involves estimating material properties or initial physical states of granular masses from observed runout patterns, or designing flow mitigating earth structures to achieve a desired outcome of the granular flows.

Existing methods for inverse analysis in granular flows face significant challenges. Conventional numerical simulators like DEM and MPM are computationally prohibitive for the repeated forward evaluations required in optimization for inverse analysis, and their non-differentiability prohibits the utilization of efficient optimization methods like gradient-based optimization. Surrogate models based on conventional machine learning or statistical models have limited generalization outside their training domain since they do not explicitly consider the granular flow dynamics or are constrained on low dimensional spaces. These issues restrict the feasibility of inverse analysis in complex granular flow problems.

In this study, we have presented an efficient framework for solving complex inverse problems in granular flows using graph neural network-based simulators (GNS). By leveraging the computational efficiency, differentiability, and generalization capabilities of GNS, coupled with gradient-based optimization through automatic differentiation, our approach successfully identifies optimal parameters to achieve desired outcomes in diverse granular flow scenarios. 

Demonstration

Inverse problems involve determining the underlying causes or parameters from observed effects or outcomes. We have demonstrated the effectiveness of our methodology across three distinct inverse analysis scenarios: (1) determining material friction angles to match target runout distances, (2) inferring multi-layered initial velocity profiles to replicate final deposit patterns, and (3) optimizing debris-resisting baffle design to control runouts. 

(1) Determining material friction angles to match target runout distances

 The objective of the inverse analysis in this section is to infer the friction angle of the granular column mass that produces a target runout distance from the friction angle of 21 degrees.

The following figure shows the optimization progress of the proposed framework. It identifies the correct friction angle (=22.45°) close to the target value (=21°) as well as the overall geometry of the final deposit. The computation time for the optimization accomplishes about 126x speed-up compared to solely relying on the high-fidelity numerical simulator owing to the computation efficiency of the GNS. 

(2) Inferring multi-layered initial velocity profiles to replicate final deposit patterns

The proposed method can be more efficient for solving multi-variable inverse problems since it uses reverse mode automatic differentiation.  Consider the multi-layered granular column which has different initial velocities for each layer (see the figure below). 

The objective of the inverse problem in this example is to estimate the initial velocity of each layer only with the information about the final deposit for the last few timesteps. The optimization results for the velocities are as follows. As the iteration increases, the velocity profile becomes similar to the true value (black line).

The animations below show the comparison between the ground truth and estimated rollout. The granular flow with the inverse estimation shows a great match with the ground truth rollout.

(3) Optimizing debris-resisting baffle design to control runouts

Our method can be advanced to the inverse design of earth structures--automatically identifies the design parameters of structures to make a system exhibit a desired outcome. We demonstrate solving the inverse design problem for the debris-resisting baffle dam array.

The figure below shows the optimization process to infer the locations of baffles that guide the centroid of the flow toe to a design location. At the final iteration, our approach successfully identifies the optimal location. 

Publications and Presentations

• Choi, Y., Kumar, K. (2024). "Inverse analysis of granular flows based on graph neural network-based simulator." Computers and Geotechnics. https://arxiv.org/abs/2401.13695

• Kumar, K., Choi, Y. (2023). “Accelerating Particle and Fluid Simulations with Differentiable Graph Networks for Solving Forward and Inverse Problems.” The International Conference for High Performance Computing, Networking, Storage and Analysis, Denver, USA. https://doi.org/10.48550/arXiv.2309.13348

• Kumar, K., Mehta, C., Choi, Y. (Oral presenter) (2023). “Graph Network Simulator and Differentiable Material Point Method.” Advances in Computational Mechanics 2023, Austin, USA.

• Choi, Y. (Oral presenter), Kumar, K. (2023). “Slope Inverse Analysis Using Accurate and Generalizable Data-Driven Surrogate Granular Flow Simulator.” 17th National Congress on Computational Mechanics, Albuquerque, USA.

• Open-source code and data at https://github.com/geoelements/gns-inverse-examples.