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

1 Introduction

• Depth-averaged Savage–Hutter equations model avalanche height and velocity while representing inertia, basal Coulomb friction, and internal Mohr–Coulomb earth pressure.Depth averaging reduces a three-dimensional problem by integrating out vertical dependence, motivated by flow thickness being much smaller than downslope travel distance.Source: §1 · p. 1
• Existing curved-terrain solvers can be accurate, but mesh generation, wet–dry-front treatment, and specialized code impose substantial setup effort.Prior work extended depth-averaged models to curved chutes and more complex terrain, alongside established numerical methods.Source: §1 · p. 2
• Earlier Savage–Hutter PINNs were limited to planar beds and constant earth-pressure coefficients, whereas curved beds require spatially varying inclination and strain-rate-dependent closure.Curved geometry also removes the closed-form solution available for planar verification, making measured experimental trajectories the basis for validation.Source: §1 · p. 2
• PINN trainability remains a practical obstacle, while the informativeness of measurement locations has received less attention than measurement quantity.The paper investigates both issues in a curved granular-flow setting.Source: §1 · pp. 2–3
• The study tests whether staged curriculum training and sparse-observation timing affect held-out trajectory accuracy in a curved-chute PINN.Its benchmark uses measured front- and rear-edge trajectories, with selected recorded times withheld from training.Source: §1 · pp. 2–3

2.1 Governing equations

• The governing system allows bed inclination to vary downslope and switches earth pressure between active and passive Mohr–Coulomb states according to local strain rate.The momentum balance retains gravity, basal Coulomb friction, and earth-pressure effects; its strain-rate-dependent closure couples the predicted velocity gradient back into the residual.Source: §2.1 · pp. 3–4

2.2 Chute geometry

• The exponentially curved chute spans x ∈ [0, 22], with flow released from rest as a unit-height parabolic pile centered at x = 1.25.Its bed angle decreases downslope through a flattening transition toward the runout region.Source: §2.2 · p. 5

2.3 Network Architecture

• A fully connected network maps space–time inputs to height and velocity, enforcing nonnegative height while leaving velocity unconstrained.The height output uses a squared softplus activation; velocity may take either sign as the mass extends or compresses.Source: §2.3 · p. 5
• The composite objective combines PDE residuals, initial and wall conditions, edge observations, and a global mass-conservation penalty.The data term constrains predicted height near zero at measured front and rear edges, rather than fitting the unobserved interior shape.Source: §2.3 · pp. 5–6
• Training first fits the initial state, then progressively widens the time window before full-domain refinement and a final L-BFGS phase.The curriculum introduces PDE and data losses over eight windows from twin = 0.5 to 14; observations contribute only after the window reaches their timestamp.Source: §2.3 · pp. 6–7

2.4 Edge extraction and evaluation metric

• Predicted edges are extracted using a height threshold, and held-out front- and rear-edge errors are combined into a trajectory evaluation metric.The field is sampled at 1,200 spatial points, with edges defined by h > 0.025; four of sixteen Experiment 73 observations are withheld for evaluation.Source: §2.4 · p. 7

3.1 Validation against experimental trajectories

• Held-out track_err was 0.77 for the eight-row reference, with front and rear RMSEs of 0.71 and 0.52, respectively.The maximum absolute error was 1.15; the front-edge error is more sensitive to the extraction threshold because the leading tail is thin.Source: §3.1 · p. 7, Figure 1 · p. 8

3.2 Effect of the staged curriculum

• Removing temporal staging raised track_err from 0.77 to 69.03, while a coarser three-stage schedule yielded 120.33 under a matched training budget.The authors attribute the sensitivity to feedback between velocity-gradient errors, the strain-rate-dependent earth-pressure coefficient, and the momentum residual.Source: §3.2 · p. 10, Table 1 · p. 10
• Removing the time-dependent PDE weight raised track_err to 2.56, removing initial-condition decay raised it to 1.20, and removing both raised it to 2.85.These ablations retain the reference eight-row budget and training schedule while removing one or both loss-scaling factors.Source: §3.2 · p. 10, Table 2 · p. 10
• Mass-conservation error remained 0.12–0.20% across the curriculum ablations despite trajectory errors differing by nearly two orders of magnitude.Because a global integral does not constrain spatial distribution, the authors identify a held-out spatial metric such as track_err as necessary for runout accuracy assessment.Source: §3.2 · pp. 10–11, Table 1 · p. 10

3.3 Effect of sparse-data placement

• With evenly spread observations, track_err stayed between 0.77 and 0.97 from eleven to six rows, then rose to 65.08 at four rows.The mass-conservation error did not indicate the sharp loss of trajectory accuracy.Source: §3.3 · p. 11, Table 3 · p. 11
• With four observations, transition-bracketing achieved track_err 0.81, near the eight-row reference 0.77; early- and late-clustered observations yielded 42.96 and 64.18.The transition-bracketing observations span t = 3.4–6.8, where downslope forcing changes sign and the flow begins to decelerate; the authors cannot separate the physical placement effect from curriculum activation duration.Source: §3.3 · pp. 11–12, Table 4 · p. 11
• The four-observation solution matched the reference bulk height to a mean absolute difference of 0.02, under 2% of peak depth averaged over t = 1–11.Its front-position RMS error was 0.81, concentrated in the thin leading tail; Figure 5 shows the transition-bracketing height field.Source: §3.3 · p. 12, Figure 5 · p. 13

• The reference configuration reproduced Experiment 73 trajectories with a held-out track_err of 0.77, using four of sixteen observations for evaluation.Source: §4 · p. 13
• Progressively expanding the training window reduced held-out error by approximately two orders of magnitude versus training over the full trajectory from the outset.The authors attribute this to feedback from the strain-rate-dependent closure, which uses the network-predicted velocity gradient.Source: §4 · p. 13
• Four observations bracketing the friction-angle transition achieved a track_err of 0.81, close to the eight-observation reference, while early- or late-only clusters performed much worse.Source: §4 · p. 13
• Mass-conservation error stayed below 0.25% even for poor trajectory predictions, so it cannot alone assess accuracy when flow distribution is the target.Source: §4 · p. 13
• Evidence for generalization is limited by sixteen observations from one experiment, and predicted height and velocity fields lack independent validation because only edge trajectories were measured.Source: §4 · p. 13
• The placement advantage remains confounded by curriculum timing, and future work proposes testing other curved geometries and separating physical placement from activation stage.The authors also propose extending the formulation to two-dimensional curved terrain.Source: §4 · p. 14