Non-local physics-informed neural networks for forward and inverse solutions of granular flows

TL;DR

A data-driven platform based on Physics-Informed Neural Networks embedded with the NGF model, capable of solving granular flows in a forward or inverse manner, demonstrates the feasibility of data-driven parameter inference in complex nonlocal models and opens up new possibilities for characterizing granular materials from sparse experimental observations.

Abstract

Dense granular flows exhibit nonlocal effects due to stress transmission in microplastic events, especially in quasi-static or slowly sheared regions. Hence, traditional local rheological models fail to capture spatial cooperativity effects that are prominent in many granular systems. The nonlocal granular fluidity (NGF) model addresses this limitation by introducing a diffusive-like partial differential equation for a fluidity field, governed by a key material-dependent parameter: the nonlocal amplitude A. However, determining A from experiments or simulations is known to be difficult and typically requires extensive calibration across multiple geometries. In this work, we present a data-driven platform based on Physics-Informed Neural Networks (PINNs) embedded with the NGF model, capable of solving granular flows in a forward or inverse manner. We show that once trained on transient flow fields, these non-local PINNs can readily infer the material parameters, as well as the pressure and stress fields. These data-driven frameworks allow for accurate recovery of small variations in the nonlocal amplitude, A, which lead to sharp bifurcation-like transitions in the flow field. This approach demonstrates the feasibility of data-driven parameter inference in complex nonlocal models and opens up new possibilities for characterizing granular materials from sparse experimental observations.

Figures (5)

• Figure 1: Schematic of the Physics-Informed Neural Network (PINN) framework developed in this work for structured fluid modeling. The architecture consists of a deep neural network coupled with embedded physical laws. Spatial and temporal coordinates $(x, y, t)$ are provided as inputs, and the network predicts physical fields including velocity $(u, v)$, pressure $P$, and stress components $(\sigma_{xy}, \sigma_{xx}, \sigma_{yy})$. Automatic differentiation is used to compute the required spatial and temporal derivatives, which are substituted into the governing equations $F_i(\sigma, P, u, v)$ representing momentum conservation, incompressible continuity, and nonlocal fluidity evolution. The resulting residuals, together with initial and boundary condition constraints, form the total loss function guiding training and enforcing physical consistency.
• Figure 2: Comparison of velocity profiles and absolute errors between numerical solutions and Direct PINN predictions for dense granular flow governed by NGF in a planar shear configuration. The top and bottom panels correspond to nonlocal amplitude values of $A = 0.97$ and $A = 1.05$, respectively. Simulations are performed with solid density $\rho_s = 2450\, \mathrm{kg/m^3}$, particle diameter $d = 0.0008\, \mathrm{m}$, no-slip boundary conditions, and zero initial velocity. The first column shows the reference velocity field obtained from a spectral numerical solver, the second column presents PINN prediction, and the third column displays the point-wise absolute error.
• Figure 3: Comparison of velocity profiles and corresponding absolute errors between forward and inverse PINN predictions for three test cases in the shear-driven configuration: $A = 0.84$ (top), $A = 0.85$ (middle), and $A = 1.03$ (bottom). The model is trained using only velocity observations and the governing equations, with no labeled stress or fluidity data. All simulations assume a two-dimensional domain with solid density $\rho = 2450~\mathrm{kg/m^3}$ and particle size $d = 0.0008~\mathrm{m}$. The first column shows the reference velocity field, the second column presents the Inverse PINN prediction, and the third column shows the point-wise absolute error.
• Figure 4: Comparison of velocity profiles and absolute errors between numerical solutions and Direct PINN predictions for dense granular flow governed by the Nonlocal Granular Fluidity (NGF) model in a pressure-driven configuration. The simulation domain is two-dimensional with no-slip frictional boundaries at the top and bottom, and flow is driven by a horizontal pressure gradient. The material parameters are density $\rho_s = 2450\, \mathrm{kg/m^3}$ and particle diameter $d = 0.0008\, \mathrm{m}$, with an initial condition of zero velocity throughout the domain. The first column shows the reference velocity field obtained from a numerical spectral solver, the second column presents the Direct PINN prediction trained with known rheological parameters, and the third column displays the pointwise absolute error. The nonlocal amplitude used in these simulations is $A = 0.48$.
• Figure 5: Comparison of velocity profiles and absolute errors from direct and inverse PINN models in a pressure-driven NGF flows. The simulation domain is two-dimensional with frictional no-slip boundaries and a horizontal pressure gradient driving the flow, using a density $\rho = 2450~\mathrm{kg/m^3}$ and particle diameter $d = 0.0008~\mathrm{m}$. The first panel shows the reference velocity field from forward PINN (trained with known rheological parameters), the second panel presents inverse PINN prediction (using only velocity observations and treating $A$ as an unknown), and the third panel displays the point-wise absolute error. The inferred value $A = 0.476$ closely matches the true value $A = 0.48$, confirming the effectiveness of the inverse framework under dynamic pressure-driven conditions.