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

This paper presents a data-driven platform using Physics-Informed Neural Networks (PINNs) embedded with the nonlocal granular fluidity (NGF) model to accurately solve forward granular flow problems and infer key material parameters, such as the nonlocal amplitude, from sparse experimental observations where traditional local models fail.

The Gist

Imagine you are watching a crowd of people at a concert. If everyone is just standing still, they act like a solid wall. If the music starts and everyone runs, they act like a fluid. But what happens in the middle? What if a few people start shoving, and suddenly, a wave of movement ripples through the crowd, even though the people at the back aren't being pushed directly?

This is exactly what happens with granular materials like sand, coffee beans, or snow. They are everywhere, from your morning cereal to the soil under your house. Scientists have been trying to write "rulebooks" (mathematical models) to predict how these materials move.

The Problem: The "Local" Rulebook Failed

For a long time, scientists used "local" rulebooks. These rules said: "If you push a grain of sand here, it only moves because of the force right next to it."

But in reality, sand is social. When you push one grain, it talks to its neighbors, who talk to their neighbors. This is called non-local behavior. It's like a rumor spreading through a school; the effect isn't just on the person who heard it first, it ripples out.

The old rulebooks couldn't handle this. They failed to predict things like:

• Shear Bands: Why does a pile of sand slide in a thin, sharp line while the rest stays still?
• Creep: Why does a mountain of sand slowly shift over years, even when it looks like it's not moving?

To fix this, scientists created a new, more complex rulebook called the Nonlocal Granular Fluidity (NGF) model. It introduces a special "social distance" parameter (let's call it A) that measures how far the "push" travels through the crowd.

The Catch: The Mystery Number

Here's the trouble: While the NGF model is great at describing reality, nobody knows the exact value of that special parameter A for a specific pile of sand.

• You can't measure it with a ruler.
• You can't weigh it.
• To find it, scientists used to have to run thousands of expensive computer simulations and tweak the number until the model matched the real world. It was like trying to guess the secret ingredient in a soup by tasting it, but you have to cook the soup from scratch every time you take a sip.

The Solution: The "Physics Detective" (PINNs)

This paper introduces a new tool called Physics-Informed Neural Networks (PINNs). Think of this as a super-smart detective that doesn't just look at clues (data); it also knows the laws of physics by heart.

The researchers built a two-step system:

1. The Simulator (Forward Mode): First, they taught the AI the rules of the game (the NGF equations). They gave it a specific "social distance" number (A) and asked, "If the sand behaves this way, what will the movement look like?" The AI predicted the flow perfectly, matching complex computer simulations.
2. The Detective (Inverse Mode): This is the magic part. They gave the AI only a video of the sand moving (the velocity). They didn't tell it the "social distance" number (A). They said, "You know the physics rules. You see how the sand is moving. Now, figure out what the secret number A must be to make this happen."

The Results: Cracking the Code

The AI didn't just guess; it solved the puzzle.

• It looked at the movement patterns and successfully calculated the hidden parameter A with incredible accuracy (less than 1% error).
• It could also figure out the pressure and stress inside the pile, things we usually can't see without breaking the pile apart.

Why This Matters (The Analogy)

Imagine you are a doctor trying to diagnose a patient.

• Old Way: You have to cut the patient open (expensive simulations) to see the internal organs, measure them, and then guess what's wrong.
• New Way (This Paper): You just listen to the patient's heartbeat and look at their skin (sparse data). Because your AI assistant knows exactly how the human body works (physics), it can tell you, "Based on this heartbeat, the patient's heart rate is exactly 72 BPM," without ever cutting them open.

The Big Picture

This research is a game-changer because:

1. It's Fast: It skips the slow, expensive trial-and-error simulations.
2. It's Smart: It works even when we only have a little bit of data (like a few video frames of sand moving).
3. It's Universal: It can be applied to any shape of container, from a narrow pipe to a giant silo.

In short, the authors have taught a computer to understand the "social life" of sand grains. By combining the laws of physics with modern AI, they can now look at a pile of sand, watch it move, and instantly know the hidden rules that govern its behavior. This opens the door to better designing everything from earthquake-resistant buildings to more efficient grain storage.

Technical Summary

1. Problem Statement

Dense granular flows exhibit complex behaviors that traditional local rheological models fail to capture, particularly in quasi-static or slowly sheared regions.

• Limitations of Local Models: Conventional models (e.g., $\mu(I)$ rheology) assume a one-to-one relationship between stress and strain rate based solely on local conditions. They break down in spatially heterogeneous deformations (e.g., creeping zones, boundary regions) where particle interactions and momentum transfer extend beyond the immediate neighborhood (nonlocal effects).
• The Nonlocal Granular Fluidity (NGF) Challenge: The NGF model addresses this by introducing a diffusive partial differential equation (PDE) for a "fluidity field" ($g$). However, this model relies on a critical, dimensionless material parameter: the nonlocal amplitude ($A$).
  • $A$ controls the spatial cooperativity length scale (how far local flow influences neighboring regions).
  • The Core Difficulty: $A$ cannot be measured directly, even in simulations, because it governs an implicit state variable (fluidity) that is not observable. Determining $A$ typically requires computationally expensive, geometry-specific calibration using Discrete Element Method (DEM) simulations and fitting procedures, limiting the model's transferability to new flow configurations.

2. Methodology

The authors propose a Physics-Informed Neural Network (PINN) framework to solve the coupled system of governing equations for dense granular flows, enabling both forward prediction and inverse parameter inference.

A. Governing Equations

The PINN is embedded with the full system of equations:

1. Conservation Laws: Incompressible continuity ($\nabla \cdot \mathbf{v} = 0$) and linear momentum conservation (including gravity body forces).
2. Nonlocal Constitutive Relation: The NGF equation, a diffusion-like PDE for the fluidity field $g$:
   $$t_0 \dot{g} = A^2 d^2 \nabla^2 g - \left[ (\mu_s - \mu)g + b \sqrt{\frac{\rho_s d^2}{P}} \mu g^2 \right]$$
   Where $A$ is the nonlocal amplitude, $d$ is particle diameter, $\mu_s$ is static yield threshold, and $b$ is rate sensitivity.
3. Stress-Strain Relation: Shear stress is related to fluidity via $\dot{\gamma} = \mu g$.

B. Neural Network Architecture

• Inputs: Spatiotemporal coordinates $(x, y, t)$.
• Outputs: Velocity components ($u, v$), pressure ($P$), stress components ($\sigma_{xx}, \sigma_{yy}, \sigma_{xy}$), and in the inverse case, the parameter $A$.
• Structure: A fully connected feedforward network with 6 hidden layers (60 neurons each) using hyperbolic tangent (tanh) activation.
• Differentiation: Automatic differentiation is used to compute spatial and temporal derivatives required for the PDE residuals, eliminating the need for mesh discretization.

C. Training Strategy

The framework employs a two-stage approach:

1. Direct PINN (Forward Problem): Trained with a prescribed value of $A$ to solve the full coupled system. The loss function minimizes residuals of the governing equations, initial conditions (IC), and boundary conditions (BC).
2. Inverse PINN (Inverse Problem): Trained using velocity observations only. The nonlocal amplitude $A$ is introduced as an additional trainable global scalar parameter within the loss function. No labeled data for stress or fluidity is required.
   • Loss Function: A composite loss ($MSE = MSE_R + \omega_2 MSE_{IC} + \omega_3 MSE_{BC}$) enforces physical consistency.
   • Optimization: Direct PINNs use a two-stage optimizer (Adam followed by L-BFGS-B); Inverse PINNs use Adam exclusively to ensure stability when optimizing network weights alongside the global parameter $A$.

D. Test Cases

The framework was validated on two canonical 2D configurations:

1. Shear-Driven Flow: Planar shear between parallel plates (top plate moving, bottom fixed).
2. Pressure-Driven Flow: Flow driven by a horizontal pressure gradient in a channel (no-slip boundaries).

3. Key Contributions

• First Practical Inverse Method for NGF: The paper presents the first data-driven framework capable of inferring the nonlocal amplitude $A$ directly from sparse velocity data without requiring internal state labels (stress/fluidity) or extensive DEM calibration.
• Mesh-Free Nonlocal Solver: It provides a unified, mesh-free approach to solving the complex, coupled nonlocal PDEs governing granular flows, overcoming the difficulties of traditional finite element/volume methods in complex geometries.
• Parameter Sensitivity Analysis: The study systematically quantifies how $A$, static friction ($\mu_s$), and rate sensitivity ($b$) influence flow localization, shear band width, and sub-yield creep.

4. Results

• Forward Accuracy: The Direct PINN accurately reproduced velocity, stress, and pressure fields for both shear-driven and pressure-driven flows, showing excellent agreement with reference solutions from a spectral basis-expansion numerical solver.
• Inverse Parameter Recovery:
  • The Inverse PINN successfully inferred the nonlocal amplitude $A$ with sub-percent relative error (typically < 0.2%) across various test cases.
  • Table I Results: For true $A$ values of 0.83, 0.84, 0.85, 1.03, and 1.05, the predicted values were 0.8307, 0.8410, 0.8498, 1.0314, and 1.0492, respectively.
  • In the pressure-driven case, the inferred $A$ (0.476) closely matched the true value (0.48).
• Flow Field Reconstruction: The inverse model accurately reconstructed the full flow evolution, including shear-band structures and transient responses, using only velocity data.
• Sensitivity: The results confirmed that the flow field is highly sensitive to small variations in $A$.
  • Higher $A$: Leads to broader shear bands, deeper fluidity penetration into sub-yield zones, and smoother velocity gradients (enhanced nonlocal cooperativity).
  • Lower $A$: Results in sharply localized deformation near the yield surface.

5. Significance

• Bridging Experiment and Theory: This work establishes a practical route to connect experimentally observable velocity fields (often the only available data) to detailed nonlocal continuum models.
• Efficiency and Generalizability: Unlike traditional calibration methods that are geometry-dependent and computationally intensive, this PINN framework generalizes across flow configurations and requires minimal data.
• Physical Interpretability: By enforcing the nonlocal PDE as a hard constraint, the model ensures that the learned parameters are physically consistent, capturing both local constitutive behavior and long-range cooperative effects (e.g., shear banding, secondary rheology).
• Future Applications: The framework is scalable and can be extended to multi-parameter inference, 3D domains, irregular geometries, and real-world experimental datasets, offering a powerful tool for characterizing granular materials in industrial and geotechnical applications.