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Physics-Informed Neural Networks for Depth-Averaged Avalanche Dynamics

Pradyumn Singh Sikarwar, Vishal Sharma, Gaurav Bhutani

arXiv:2609.34916v1physics.flu-dyncond-mat.softcs.LG

TL;DR

Avalanche motion is hazardous to mountain communities, yet its depth-averaged equations pose PINN training challenges from frictional terms and moving wet–dry fronts. This study develops a PINN for the Savage–Hutter model and tests it from one-dimensional analytical verification to two-dimensional experimental validation. The coupled one-dimensional model reconstructs both fields, while sparse final-deposit data support two-dimensional training and consistent accuracy across the reported variations in pile mass and slope angle.

  • Problem

    The study asks whether PINNs can solve depth-averaged avalanche equations relevant to hazard assessment despite non-smooth friction, sign-dependent source terms, and moving wet–dry fronts that complicate training.

    Source: §1 · pp. 1–3

  • Method

    The authors train PINNs for the Savage–Hutter equations, verifying one-dimensional predictions analytically and extending two-dimensional training with sparse final-deposit observations.

    Source: §6 · p. 28

  • Results

    Mean height and velocity RMSEs were 4.3×10−2 and 7.9×10−2, respectively, for the coupled one-dimensional conservative formulation; sparse-data two-dimensional accuracy was consistent across pile-mass and slope-angle variations.

    Source: §6 · p. 28

  • Takeaways & Limitations

    Ten sparse final-deposit measurements per case, combined with governing equations, were sufficient to reconstruct full spatio-temporal flow histories consistently across the reported pile-mass and slope-angle variations.

    Source: §6 · p. 28

    The framework requires at least sparse final-deposit measurements, and its approximately 6.5-hour training time per case exceeds a single TITAN2D simulation.

    Source: §6 · p. 29

Abstract

from arXiv · show

Accurate prediction of avalanche motion is essential for hazard assessment in mountainous terrain. This study develops and evaluates a physics-informed neural network (PINN) framework for the Savage-Hutter model of depth-averaged granular flow, progressing from 1D analytical verification to 2D experimental validation. First, three 1D problems of increasing complexity were verified against the analytical solution: height prediction with prescribed velocity, velocity prediction with prescribed height, and coupled prediction of both fields using the conservative formulation. The decoupled tests accurately reconstructed the spatio-temporal evolution of each field when the other was prescribed. The coupled formulation learned both fields without prescribed data, achieving mean height and velocity RMSEs of 0.043 and 0.079 in non-dimensional units. A hyperparameter sensitivity study evaluated the effects of network depth, width, collocation density, learning rate, and epochs. The framework was then extended to 2D and validated against laboratory experiments of a cylindrical granular pile collapsing on an inclined plane, with TITAN2D providing numerical comparisons. Purely physics-based training converged to the trivial zero solution; augmenting the loss with 10 sparse training points from final deposit profiles produced a physics-informed, data-assisted hybrid framework. Peak flow depth, depth-averaged velocity, RMSE, and wetted-area IoU evaluated global and local agreement. Global height RMSE ranged from 2.7 to 6.7 mm across four experimental cases, while mean wetted-area IoU ranged from 69 to 81 %, demonstrating consistent performance across variations in pile mass and slope angle.

1 Introduction

The study introduces PINNs for depth-averaged granular avalanche equations, a setting with distinct frictional and slope-dependent physics. It addresses training difficulties arising from these equations and the moving wet–dry interface.

1 Introduction

  • Rapid granular avalanches threaten mountain communities and infrastructure, while full-scale experiments are dangerous, expensive, and rarely repeatable.These conditions make physics-based modelling a primary means of studying flow dynamics and assessing risk.Source: §1 · p. 1
  • Mesh-based solvers can be difficult to apply because discretisation choices, complex-terrain mesh generation, and code-development effort add setup overhead.TITAN2D is a widely used adaptive-mesh solver for depth-averaged granular flows over natural terrain.Source: §1 · p. 2
  • PINNs encode governing equations, initial and boundary conditions, and available observations in a neural-network loss, with derivatives computed by automatic differentiation.They provide a mesh-free alternative to domain discretisation.Source: §1 · p. 2
  • PINNs had been applied to hydraulic shallow-water equations, while granular-flow PINNs had used continuum-level rheological models rather than depth-averaged equations.The paper situates its contribution between these two areas.Source: §1 · p. 2
  • The study presents the first application of PINNs to depth-averaged granular avalanche equations, which include Coulomb basal friction and slope-dependent source terms.Earlier PINN work in granular flows used continuum-level rheological models rather than depth-averaged formulations.Source: §1 · p. 2
  • Near moving wet–dry fronts, sharp depth transitions and non-smooth friction and velocity-normalisation terms can produce poorly conditioned residuals and destabilise training.The system also has direction-dependent Coulomb friction and sign-dependent Mohr–Coulomb source terms.Source: §1 · pp. 2–3
  • The study evaluates a two-stage approach: one-dimensional verification against an analytical solution, followed by two-dimensional validation against laboratory experiments and TITAN2D.The two-dimensional stage uses sparse experimental observations to address the trivial-solution pathology.Source: §1 · p. 3

2 Mathematical Formulation

The formulation treats granular avalanches as a depth-averaged continuum and derives one- and two-dimensional Savage–Hutter systems. A similarity solution supplies the one-dimensional verification benchmark.

2 Mathematical Formulation

  • The continuum assumption holds bulk density constant in space and time and treats granular dilation as negligible.It is adopted for dense flows where particles remain in contact and packing variations are minimal.Source: §2 · p. 3

2.1 One-dimensional depth-averaged equations

  • Depth averaging applies when flow thickness is small relative to downslope length; the model imposes free-surface conditions and bed no-penetration with Coulomb friction.Vertical velocity and acceleration are neglected, and vertical normal stress is hydrostatic.Source: §2.1 · pp. 3–4
  • The one-dimensional system couples depth and downslope velocity through continuity and momentum equations, with hydrostatic pressure flux and gravitational and basal-friction source terms.The sign function makes basal friction oppose motion; the conservative form is described as suitable for PINN solvers.Source: §2.1 · p. 4

2.1.1 Non-dimensionalization

  • The non-dimensionalization separates flow thickness from downslope length using aspect ratio ϵ = H/L ≪ 1, leaving slope angle, friction angle, and aspect ratio as governing parameters.The aspect ratio appears in the horizontal normal-stress flux coefficient.Source: §2.1.1 · pp. 4–5

2.1.2 Analytical similarity solution

  • The one-dimensional benchmark uses a self-similar solution for rigid-body translation and lateral spreading of an initially parabolic pile.Its effective downslope acceleration is ˆa = sin ζ − cos ζ tan δ, and the time-varying half-width ˆg(ˆt) is reconstructed numerically because its defining relation has no closed-form inversion.Source: §2.1.2 · pp. 5–6

2.2 Two-dimensional depth-averaged equations

  • The two-dimensional extension adds cross-slope variation and velocity components to continuity and momentum, with curvature corrections vanishing for the planar inclined-plane geometry.The same dimensional equations are solved by TITAN2D and the PINN framework.Source: §2.2 · pp. 6–7
  • In the two-dimensional coordinate system, x points downslope, y cross-slope, and h, u, and v denote flow depth and the corresponding depth-averaged velocity components.The formulation uses the same κap = 1 normal-stress relation in both horizontal directions.Source: §2.2 · p. 6

3 Physics-Informed Neural Network Framework

The PINN represents the depth-averaged fields with a fully connected network and minimizes a composite objective built from governing-equation and condition residuals. In two dimensions, sparse depth observations supplement the physics loss to avoid the trivial zero solution.

3 Physics-Informed Neural Network Framework

  • A fully connected feed-forward network predicts the solution fields, and automatic differentiation supplies derivatives without a mesh or finite-difference stencil.Inputs map to (ˆh, ˆu) in one dimension and (h, u, v) in two dimensions; the two-dimensional depth output uses softplus to enforce non-negativity.Source: §3 · pp. 7–8
  • The PDE residual loss enforces conservation laws at interior collocation points, while initial- and boundary-condition losses enforce prescribed values at their sampled points.The two-dimensional system uses three residuals, and conservative flux terms are differentiated as composite expressions to retain coupling between depth and velocity.Source: §3 · pp. 8–9
  • In two dimensions, the zero fields satisfy the governing equations and minimize the PDE loss, so a sparse data loss is added to anchor training.The data loss compares observed depths with network predictions.Source: §3 · p. 9
  • The total objective weights PDE, initial-condition, boundary-condition, and data losses; one-dimensional verification sets the data weight to zero, while two-dimensional training uses a positive data weight.The two-dimensional optimizer runs Adam followed by up to 1000 L-BFGS steps.Source: §3 · p. 9
  • Two-dimensional PDE collocation times follow a Beta(1, 3) distribution to concentrate samples near the initial steep gradients, and points are resampled every 500 Adam epochs.A learning-rate scheduler halves the rate after 5000 consecutive epochs without a decrease in total loss.Source: §3 · p. 9

4 One-Dimensional Analytical Verification

The one-dimensional verification progresses from tests that prescribe one field to a coupled conservative formulation that learns both height and velocity. The PINN tracked analytical profiles across the simulation window, with the coupled test achieving mean RMSEs of 4.3 × 10−2 for height and 7.9 × 10−2 for velocity.

4.1 Problem configurations

  • Three tests used a parabolic pile at rest: height prediction with prescribed analytical velocity, velocity prediction with prescribed analytical height, and coupled learning with neither field prescribed.The simulations spanned ˆx ∈ [−8, 10] and ˆt ∈ [0, 4], with homogeneous Dirichlet conditions at spatial boundaries.Source: §4.1 · pp. 9–10, Table 1 · p. 10

4.2 Decoupled verification

  • The decoupled tests reproduced the prescribed-field analytical profiles, while velocity errors clustered at sharp pile edges and tapered smoothly beyond the analytical support.Height RMSE peaked at roughly 0.020 and averaged 7.2 × 10−3; velocity RMSE peaked near 0.20 and settled at a higher level.Source: §4.2 · pp. 10–12, Figure 3 · p. 11, Figure 4 · p. 12
  • The early RMSE rise in both decoupled tests reflects sharp initial pile-edge discontinuities and localized flux divergence near the compact front, with both difficulties easing as the pile spreads.The velocity test used 32 neurons versus 8 for height and five times as many collocation points.Source: §4.2 · p. 12

4.3 Coupled verification

  • A single network learned height and velocity simultaneously from the conservative equations, requiring a wider, more densely sampled network than either decoupled test.All three loss components decreased over 20,000 epochs, with LPDE and LIC reaching O(10−5–10−6).Source: §4.3 · pp. 13, 15
  • Mean RMSEs were 4.3 × 10−2 for height and 7.9 × 10−2 for velocity in the coupled formulation, which tracked analytical height and velocity profiles across the simulation window.The height error settled below 0.025 after peaking near ˆt ≈ 0.3–0.5; errors were concentrated near pile peaks early and at velocity-profile edges.Source: §4.3 · pp. 13–15, Figure 6 · p. 14, Figure 7 · p. 15

5 Two-Dimensional Validation: Experimental Granular Avalanche

The 2D PINN required sparse final-deposit observations to avoid the trivial zero solution, then reproduced experimental deposit profiles and flow diagnostics across four cases. Across cases, global height RMSE ranged from 2.7 to 6.7 mm and mean wetted-area IoU from 69.0% to 80.7%.

5.1 Experimental configuration

  • The four laboratory cases varied pile mass and slope angle, and C4 (2.5 kg, 15°) was selected as the primary, most demanding test.The experimental setup used a cylindrical dry-sand pile released on an inclined surface; all cases shared the reported material properties and initial radius.Source: §5.1 · pp. 16–17, Table 2 · p. 16, Figure 9 · p. 17

5.2 Trivial-solution pathology in two-dimensional PINNs

  • With purely physics-based losses, the network converged to h ≡ 0, u ≡ 0, v ≡ 0; adding sparse experimental observations pulled training away from this trivial solution.The zero fields satisfy the homogeneous Savage–Hutter equations, and most collocation points initially lie outside the wetted region.Source: §5.2 · pp. 17–18

5.3 Minimum-data requirement study

  • Five observations per centerline and cross-slope profile (10 total) were adopted after global height RMSE improved sharply through N = 3 and changed little beyond N = 5.The selected observations span the deposit peak, upstream face, downstream tail, and lateral extent.Source: §5.3 · p. 18, Figure 10 · p. 18

5.4 Network architecture and two-stage training

  • Training used 500,000 Adam epochs followed by up to 1,000 L-BFGS steps, with data-loss weight 50; each PINN case took about 6.5 h versus 3–12 min for TITAN2D.A preliminary C4-to-C2 transfer-learning run reduced training time by approximately 94%, though demonstrated for only one case pair.Source: §5.4 · p. 19, Table 4 · p. 20

5.5 Results: primary case C4

  • Across C4 snapshots, the PINN tracked collapse, lateral spreading, and deposit profiles closely, while its final downslope extent exceeded TITAN2D and its wetted area was larger.The final centerline profile reproduced the experimental peak and tail, while TITAN2D slightly underpredicted both; the comparison includes Maeno et al. [2013] measurements.Source: §5.5 · pp. 21–23, Figure 13 · p. 22, Figure 14 · p. 23
  • For C4, global depth RMSE was 6.7 mm and time-averaged wetted-area IoU was 69.0%, using a 4.3 mm depth threshold for quantitative wetted-region metrics.The global RMSE averaged over the full simulation window was 3% of the initial pile height.Source: §5.5 · p. 24, Figure 17 · p. 25

5.6 Multi-case validation

  • Global RMSE was 2.7–3.1 mm for 1 kg cases and 4.7–6.7 mm for 2.5 kg cases; mean IoU ranged from 69.0% to 80.7%.Table 5 reports the four-case metrics; RMSE peaked during early collapse and decreased as deposits settled.Source: Table 5 · p. 27, §5.6 · p. 27
  • PINN wetted area exceeded TITAN2D in every case by 12.1%–32.4%, with overlap covering most of the TITAN2D footprint and discrepancies concentrated at the wet–dry boundary.The excess increased with mass at a fixed slope angle.Source: §5.6 · p. 27, Table 6 · p. 28, Figure 21 · p. 28

6 Conclusions

The study reports 1D analytical verification and 2D experimental validation of PINNs for Savage–Hutter avalanche equations, with sparse final-deposit data needed for the 2D cases. It concludes that the approach reconstructs flow histories across the tested conditions, while identifying interface, data, and computational limitations.

6 Conclusions

  • The fully coupled 1D conservative formulation achieved mean height and velocity RMSEs of 4.3×10^-2 and 7.9×10^-2 in non-dimensional units against the analytical parabolic-cap solution.The coupled fields were learned without prescribed data.Source: §6 · p. 28
  • In 2D, ten sparse final-deposit observations per case anchored training away from the zero solution while the PDE constraint preserved physical consistency.The authors conclude that final deposits combined with governing equations can reconstruct spatio-temporal flow histories across the tested mass and slope variations.Source: §6 · p. 28
  • The fully connected network smooths the wet–dry interface and overestimates wetted area; the framework also requires sparse final-deposit data and takes about 6.5 h per case.The authors identify irregular terrain, generalisability, and computational efficiency as directions for future work.Source: §6 · pp. 28–29

Computer Code Availability

The paper provides a public source-code repository and an archived TITAN2D reference dataset.

Computer Code Availability

  • Source code is available at https://github.com/Prasik3182/PINNs_Avalanche.git, and TITAN2D benchmark data are archived on Zenodo.The passage provides the dataset DOI as 10.5281/zenodo.21393644.Source: Computer Code Availability · p. 29

A Hyperparameter Sensitivity: Decoupled and Coupled Case

The sensitivity sweeps show that velocity prediction accounts for most accuracy variation, while the pseudo-2D appendix configures a strip-flow test to examine dry-region dominance without sparse data augmentation.

A Hyperparameter Sensitivity: Decoupled and Coupled Case

  • In the decoupled tests, height RMSE was insensitive to the five hyperparameters, while velocity RMSE accounted for most variation; collocation density had no systematic accuracy effect.The learning rate α = 10^-3 lay in a flat region before velocity error rose sharply at α = 10^-1.Source: A Hyperparameter Sensitivity:… · p. 31, Figure 22 · p. 32
  • For the coupled formulation, RMSEh stayed nearly flat across settings, velocity error accounted for almost all variation, and training cost scaled linearly with network size and collocation density.The selected architecture had five layers with 64 neurons per layer.Source: A Hyperparameter Sensitivity:… · p. 32, Figure 23 · p. 33

B Pseudo-2D Strip Flow: Diagnosing Dry-Region Dominance

  • The pseudo-2D strip-flow test replaced the cylindrical pile with a full-width ridge to increase the active-flow share of the domain and isolate dry-region dominance.The initial condition and flow were uniform in y, making the dynamics effectively one-dimensional despite two-dimensional inputs and governing equations.Source: B Pseudo-2D Strip Flow: Diagnosing… · p. 34

B.1 Problem configuration

  • The no-data pseudo-2D experiment used a 0.085 m ridge, the main-text network and training setup, and slope angles of 15°, 20°, and 30° over 0.55 s.Predicted-to-initial total volume was tracked as the primary mass-retention diagnostic.Source: §B.1 · p. 34

B.2 Results

In the pseudo-2D tests, training recovered from brief collapses toward the zero solution, with retained volume increasing at steeper slopes. Across the three slopes, PINN profiles remained close to TITAN2D; Case C1 diagnostics also show close tracking of maximum height.

B.2 Results

  • 96.7 % of initial volume was retained at 30◦, while the 15◦ case settled near 0.675 and was still slowly increasing at training’s end.The final volume ratios were averaged over the final 10000 epochs.Source: §B.2 · p. 34, Table 7 · p. 34
  • Across all three slope angles, training briefly collapsed toward zero near epochs 12000–14000 before recovering to a physical solution.The cross-slope profiles remained uniform in y, consistent with a strip initial condition without cross-slope gravity.Source: §B.2 · p. 34
  • At 15◦, the flow decelerated as its centreline peak advanced downslope and flattened, while wavefront position and peak height stayed close to TITAN2D.The depth contours show an elongated deposit in the downslope direction.Source: §B.2 · p. 35, Figure 24 · p. 35, Figure 25 · p. 36
  • At 20◦, reduced friction produced faster collapse and greater runout than at 15◦; the wavefront remained close to TITAN2D.The downstream tail extended farther and the deposit was more elongated than at 15◦.Source: §B.2 · p. 36, Figure 26 · p. 37, Figure 27 · p. 37
  • At 30◦, the slope exceeded the bed friction angle, and the flow accelerated while remaining close to TITAN2D throughout the simulation.The wavefront advanced rapidly, with the downstream tail extending beyond the initial pile location by t = 0.55 s.Source: §B.2 · p. 38, Figure 28 · p. 38, Figure 29 · p. 39

C.1 Case C1: 1 kg, 10◦

  • In Case C1, maximum height decayed monotonically from H0 = 85 mm, with the PINN closely tracking TITAN2D; time-mean IoU was 70.9 %.The PINN peak was slightly taller and shifted downslope relative to TITAN2D at t = 0.70 s.Source: §C.1 · pp. 39–41, Figure 30 · p. 40, Figure 31 · p. 41

C.2 Case C2: 1 kg, 15◦

The C2 and C3 cases show that the PINN reproduced key profile and integral-diagnostic behavior, with different agreement patterns across cases. Three independent C4 runs clustered closely, and their final-deposit spread was smaller than the gap to TITAN2D.

C.2 Case C2: 1 kg, 15◦

C.3 Case C3: 2.5 kg, 10◦

  • In Case C3, the PINN matched TITAN2D most closely at the final snapshot, after the larger-mass deposit had thinned and lateral gradients relaxed.Its mean-velocity peak was lower and later, and its motion persisted over a longer runout distance than TITAN2D.Source: §C.3 · pp. 43–44, Figure 36 · p. 44, Figure 37 · p. 45

D.1 Profile variability across runs

D.2 Final deposit comparison

  • At the C4 final deposit, inter-run spread was substantially smaller than the gap between the PINN band and TITAN2D in both profile directions.This supports interpreting the reported discrepancy as model behavior rather than sensitivity to random initialization.Source: §D.2 · p. 48, Figure 40 · p. 48

D.3 Error metrics across runs

Across three C4 training runs, RMSE and IoU followed similar trajectories; final RMSE stayed within a narrow band, while time-mean IoU varied by only a few percentage points.

D.3 Error metrics across runs

  • Across three C4 runs, RMSE peaked during early collapse and decayed as the deposit thinned, while IoU varied more during spreading before converging to a similar final range.Final-snapshot RMSE remained within a narrow band, and time-mean IoU varied by only a few percentage points.Source: §D.3 · p. 48, Figure 41 · p. 48

D.4 Training data point coordinates

The ten C4 training points sampled the deposit’s centerline and cross-slope extent.

D.4 Training data point coordinates

  • Ten C4 training points sampled the upstream face, deposit peak, and downstream tail along the centerline, plus the deposit’s full lateral extent along the cross-slope profile.Table 8 reports their coordinates in centimetres.Source: §D.4 · p. 49, Table 8 · p. 49
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