cond-mat.softSep 2, 2026

Physics-Informed Neural Networks for Depth-Averaged Granular Avalanche Dynamics on Curved Topography

Authors: Pujan Pranavkumar Purohit, Pradyumn Singh Sikarwar, Vishal Sharma, Gaurav Bhutani

Organizations: School of Mechanical and Materials Engineering, Indian Institute of Technology Mandi, Himachal Pradesh 175075, India

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Abstract

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving governing equations, but their application to granular avalanche dynamics over curved terrain remains largely unexplored. This study extends a depth-averaged PINN formulation based on the Savage-Hutter equations to an exponentially curved chute with spatially varying inclination and a strain-rate-dependent Mohr-Coulomb earth-pressure closure. The model is validated against measured front- and rear-edge trajectories from a laboratory granular-avalanche experiment, with selected observations withheld from training. A staged temporal curriculum proved essential for accurate prediction, reducing the held-out trajectory error by approximately two orders of magnitude compared with training over the full time domain from the outset. Sparse-data experiments further showed that observation placement was more influential than observation number within the configurations tested. Four observations bracketing the transition from acceleration to deceleration achieved nearly the same accuracy as the eight-observation reference configuration, whereas observations clustered at early or late times performed poorly. The results demonstrate the importance of both training strategy and informative data placement when applying PINNs to granular flows over curved topography.

Figures & tables

Figure 1: Predicted front and rear edge trajectories for the reference configuration (eight training rows) against Experiment 73 of Hutter and Koch [ 2 ] . Stars mark the four held-out rows, which were withheld entirely from training. Held-out track_err=0.770\text{track\_err}=0.770 .
Figure 2: Predicted height field on the curved chute geometry for the reference configuration, shown at t=0t=0 , 0.50.5 , 11 , 22 , 33 , 55 , 88 , 1111 and 1414 . The bed profile is overlaid for reference.
Figure 3: Evolution of the five loss components ( LPDE\mathcal{L}_{\mathrm{PDE}} , LIC\mathcal{L}_{\mathrm{IC}} , Lwall\mathcal{L}_{\mathrm{wall}} , Ldata\mathcal{L}_{\mathrm{data}} , Lmass\mathcal{L}_{\mathrm{mass}} ) against training step (log scale) for the reference configuration. Vertical dashed lines mark the boundaries between the four training phases described in Section 2.3 , numbered sequentially; the arrow indicates the brief final L-BFGS phase at the end of training.
Configurationtrack_err (held-out)Mass-conservation error
Eight-stage curriculum (adopted)0.770.16 %
No staging (single window)69.030.12 %
Coarse three-stage curriculum120.330.20 %
Table 1: Curriculum ablation. All three configurations use the same eight-row training budget, the same four held-out rows, and a matched total step count; only the curriculum schedule differs. Mass-conservation error is essentially unaffected by the choice of schedule, as expected of a global integral, while the trajectory metric degrades by roughly two orders of magnitude without staging.
Configurationtrack_err (held-out)Mass-conservation error
Both scaling factors (adopted)0.770.16 %
No PDE time-weight2.560.12 %
No IC decay1.200.13 %
Neither factor2.850.18 %
Table 2: Loss-scaling ablation, on the same eight-row training budget and full training schedule as Table 1 and otherwise identical settings. The reference configuration applies both the time-increasing PDE weight and the curriculum-window decay on the initial-condition terms; each ablation removes one or both.
Training rows (of 16)Held-out rowsUnused rowstrack_errMass-cons. error
11410.850.14 %
10420.850.16 %
8 (reference)440.770.16 %
6460.970.15 %
4 (evenly spaced)4865.080.11 %
Table 3: Training-data budget against held-out accuracy, using rows spread evenly across the record at each step, ordered from most to fewest. Accuracy remains nearly unchanged from eleven to six rows but deteriorates sharply between six and four rows, while the mass-conservation error provides no indication of the failure.
PlacementTimes used (non-dim.)track_err (held-out)
Clustered late9.3, 10.1, 11.0, 12.464.18
Clustered early0.9, 2.5, 3.4, 4.242.96
Transition-bracketing3.4, 4.2, 6.0, 6.80.81
Table 4: Sparse-data placement study. All three configurations use exactly four training rows and the same four held-out rows as in Tables 1 – 3 ; only the placement of the four training rows differs. The eight-row reference configuration in Table 3 gives track_err=0.77\text{track\_err}=0.77 for comparison.
Figure 4: Predicted trajectories for the transition-bracketing configuration using four training rows. The dotted segment beyond t≈13t\approx 13 lies outside the evaluation window and reflects domain truncation rather than physical arrest.
Figure 5: Predicted height field for the transition-bracketing configuration at the same time instants as Figure 2 .

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