Physics-Informed Neural Networks for Depth-Averaged Avalanche Dynamics

Pradyumn Singh Sikarwar; Vishal Sharma; Gaurav Bhutani

physics.flu-dynSep 28, 2026

Physics-Informed Neural Networks for Depth-Averaged Avalanche Dynamics

Authors: Pradyumn Singh Sikarwar, Vishal Sharma, Gaurav Bhutani

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

Abstract

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.

Figures & tables

Figure 1 : Geometric configuration of the one-dimensional benchmark problem. The initial pile follows the parabolic non-dimensional profile on an inclined plane of angle

\zeta

. The front and rear positions are denoted by

\hat{x}_{F}

and

\hat{x}_{R}

Figure 2 : Schematic of the PINN framework used in this study.

Network architecture
Hyperparameter	Decoupled- $h$	Decoupled- $u$	Coupled
Output variables	$h(x,t)$	$u(x,t)$	$h(x,t)$ , $u(x,t)$
Width (Neurons per layer)	8	32	64
Depth (Hidden layers)	5	5	5
Activation	$\tanh$	$\tanh$	$\tanh$
Weight initialisation	Xavier uniform	Xavier uniform	Xavier uniform

Table 1 : Hyperparameter settings for the one-dimensional decoupled and coupled formulations.

Panel of Figure 3 — Figure 3 : Decoupled- $h$ : analytical and PINN-predicted height profiles at six time instants.

Panel of Figure 4 — Figure 4 : Decoupled- $u$ : analytical and PINN-predicted velocity profiles at six time instants.

Panel of Figure 5 — Figure 5 : Decoupled verification: temporal evolution of RMSE for the height field (a) and velocity field (b).

Figure 6 : Coupled formulation: analytical and PINN-predicted height profiles at

\hat{t}=0

0.5

1.0

2.0

3.0

, and

4.0

Figure 7 : Coupled formulation: analytical and PINN-predicted velocity profiles at

\hat{t}=0

0.5

1.0

2.0

3.0

, and

4.0

. The shaded band marks the active region.

Panel of Figure 8 — Figure 8 : Coupled PINN training diagnostics: (a) component losses, (b) final-epoch weight histogram ( $\mu=0.000$ , $\sigma=0.149$ ), and (c) RMSE of height and velocity fields over time.

Case	$M$ ( $\mathrm{kg}$ )	$\zeta$ ( $\mathrm{\SIUnitSymbolDegree}$ )	$H_{0}$ ( $\mathrm{m}$ )	$t_{\max}$ ( $\mathrm{s}$ )
C1	$1.0$	$10$	$0.085$	$0.70$
C2	$1.0$	$15$	$0.085$	$0.70$
C3	$2.5$	$10$	$0.215$	$0.95$
C4	$2.5$	$15$	$0.215$	$0.95$

Table 2 : Experimental cases from Maeno et al. (2013) were used to validate the two-dimensional PINN. All cases share

R_{0}=

0.05\text{,}\mathrm{m}

, $\rho=$1481\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}

\phi=24^{\circ}

, and

\delta=28^{\circ}

Panel of Figure 9 — Figure 9 : Three-dimensional schematic of the experimental configuration on a plane inclined at angle $\zeta$ to the horizontal, with $x$ directed down-slope, $y$ cross-slope, and $z$ normal to the inclined surface. (a) Initial condition: a cylindrical granular pile of radius $R_{0}$ and height $H_{0}$ centred at the origin. (b) Final deposit: the collapsed pile spreads into an elongated elliptical deposit with greater runout in the downslope direction.

Figure 10 : Global height RMSE as a function of the number of observations per profile,

N

, for C4 (

2.5\,\text{kg}

15^{\circ}

), compared with TITAN2D (orange circles) and experimental measurements (blue squares).

Property	Value
Input dimension	3 ( $x$ , $y$ , $t$ )
Hidden layers	5
Neurons per layer	64
Activation	$\tanh$
Output activation on $h$	softplus
Weight initialisation	Xavier uniform

Table 3 : 2D PINN architecture and training configuration for the Maeno et al. (2013) cases.

Case	Configuration	PINN training time (h)	TITAN2D simulation time (min)
C1	$1.0\,\text{kg}$ , $10^{\circ}$	$\approx 6.5$	3
C2	$1.0\,\text{kg}$ , $15^{\circ}$	$\approx 6.5$	7
C3	$2.5\,\text{kg}$ , $10^{\circ}$	$\approx 6.5$	6
C4	$2.5\,\text{kg}$ , $15^{\circ}$	$\approx 6.5$	12

Table 4 : Wall-clock training and simulation times for all four experimental cases. PINN training was performed on an NVIDIA RTX A5000 GPU; TITAN2D simulations were run on 8 CPU cores. The PINN time is identical across cases because training duration depends solely on the machine-learning parameters, which are held fixed throughout.

Panel of Figure 11 — Figure 11 : Training loss history for C4. (a) Adam stage: all loss components decreased monotonically; the data loss dominated early training and guided the network away from the trivial solution. (b) L-BFGS stage: rapid convergence within $\approx 50$ steps, reducing $\mathcal{L}_{\text{total}}$ from $6.0\times 10^{-4}$ to $4.5\times 10^{-4}$ .

Panel of Figure 12 — Figure 12 : Centerline profile $h(x,\,y{=}0,\,t)$ for C4 at six training checkpoints. PINN (solid), TITAN2D (dashed) at $t=0$ , $0.32$ , $0.63$ , $0.95\,\text{s}$ .

Panel of Figure 13 — Figure 13 : Centerline (left) and cross-slope (right) height profiles at four time snapshots for C4. PINN (solid blue) and TITAN2D (dashed orange).

Panel of Figure 14 — Figure 14 : Final deposit profiles at $t=0.95\,\text{s}$ for C4. Left: centerline ( $y=0$ ); right: cross-slope ( $x=0$ ). PINN (solid blue), TITAN2D (dashed red), and Maeno et al. (2013) experimental data (circles). Filled red circles indicate the training data points.

Panel of Figure 15 — Figure 15 : Depth contours for C4 at $t=0.00$ , $0.23$ , $0.47$ , and $0.95\,\text{s}$ (columns, left to right). Top row: PINN; bottom row: TITAN2D . Contour levels are identical across columns. The grey region marks the wetted footprint, defined as the area where the flow depth exceeds the minimum threshold.

Panel of Figure 16 — Figure 16 : Integral flow diagnostics for C4: $h_{\max}$ and $\bar{V}$ versus time (top row) and runout distance (bottom row). TITAN2D (solid red); PINN (dashed blue).

Panel of Figure 17 — Figure 17 : Time-resolved error metrics for C4. Left: RMSE of the depth field $h$ ; the global value averaged over all time steps is $6.7\times 10^{-3}\,\text{m}$ . Right: Intersection over Union (IoU) of the wetted area; the time-averaged value is $69.0\,\%$ .

Figure 18 : Final wetted footprints at $t=0.95\,\text{s}$ for C4. Blue: PINN ( $A_{\text{PINN}}=0.2091\,\text{m}^{2}$ ); red: TITAN2D ( $A_{\text{Titan}}=0.1579\,\text{m}^{2}$ ); purple: overlap region. The black circle marks the initial pile boundary. Wet cells were classified using the threshold $h_{\mathrm{thr}}=H_{0}/50=4.3\,\text{mm}$ .

Panel of Figure 19 — Figure 19 : Final deposit profiles for Cases C1–C3. Left column: centerline ( $y=0$ ); right column: cross-slope ( $x=0$ ). Rows from top to bottom: C1 ( $1\,\text{kg}$ $10^{\circ}$ ), C2 ( $1\,\text{kg}$ $15^{\circ}$ ), and C3 ( $2.5\,\text{kg}$ $10^{\circ}$ ). PINN (solid blue), TITAN2D (dashed red), and experimental data from Maeno et al. (2013) (circles). Filled red circles indicate training observations.

Panel of Figure 20 — Figure 20 : PINN depth contours at $t=0.20$ , $0.40$ , and $0.70\,\text{s}$ (columns, left to right) for Cases C1–C3 (rows, top to bottom): $1\,\text{kg}$ $10^{\circ}$ , $1\,\text{kg}$ $15^{\circ}$ , and $2.5\,\text{kg}$ $10^{\circ}$ . The grey region marks the wetted footprint, defined using the same height cutoff of $0.05\,\text{cm}$ used for C4.

Case	Configuration	Global RMSE (m)	Mean IoU (%)
C1	$1.0\,\text{kg}$ , $10^{\circ}$	$3.1\times 10^{-3}$	70.9
C2	$1.0\,\text{kg}$ , $15^{\circ}$	$2.7\times 10^{-3}$	80.7
C3	$2.5\,\text{kg}$ , $10^{\circ}$	$4.7\times 10^{-3}$	73.1
C4	$2.5\,\text{kg}$ , $15^{\circ}$	$6.7\times 10^{-3}$	69.0

Table 5 : Global RMSE and mean IoU for all four experimental cases.

Panel of Figure 21 — Figure 21 : Final wetted footprints for Cases C1–C3 (left to right): $1\,\text{kg}$ $10^{\circ}$ , $1\,\text{kg}$ $15^{\circ}$ , and $2.5\,\text{kg}$ $10^{\circ}$ . PINN (blue), TITAN2D (red), and overlap (purple). The black circle marks the initial pile boundary. Wetted areas are annotated on each panel. Wet cells were classified using $h_{\mathrm{thr}}=H_{0}/50$ , giving $1.7\,\text{mm}$ for C1–C2 and $4.3\,\text{mm}$ for C3.

Case	Configuration	$A_{\text{PINN}}$ (m 2 )	$A_{\texttt{TITAN2D}}$ (m 2 )	Excess (%)
C1	$1.0\,\text{kg}$ , $10^{\circ}$	0.1191	0.0965	23.4
C2	$1.0\,\text{kg}$ , $15^{\circ}$	0.1190	0.1061	12.1
C3	$2.5\,\text{kg}$ , $10^{\circ}$	0.1967	0.1538	27.9
C4	$2.5\,\text{kg}$ , $15^{\circ}$	0.2091	0.1579	32.4

Table 6 : Final wetted areas for all four experimental cases.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 22 : Hyperparameter sensitivity of the decoupled one-dimensional PINN: prediction RMSE (left) and wall-clock training time (right). Rows (a)–(e): learning rate

\alpha

, width

N_{w}

, depth

N_{d}

, collocation points

N_{\mathrm{col}}

, and epochs

N_{e}

. Error bars show mean

\pm

standard deviation over three independent runs. Height (blue circles); velocity (orange squares).

Panel of Figure 23 — Figure 23 : Hyperparameter sensitivity of the coupled one-dimensional PINN: prediction accuracy (left) and wall-clock training time (right). Rows (a)–(e): learning rate $\alpha$ , width $N_{w}$ , depth $N_{d}$ , collocation points $N_{c}$ , and epochs $N_{e}$ .

$\zeta$	Final volume ratio	Mean (last 10000 epochs)
$15^{\circ}$	0.675	0.645
$20^{\circ}$	0.745	0.723
$30^{\circ}$	0.967	0.969

Appendix

Table 7 : Final volume ratios for the pseudo-2D experiment. Values are reported at the end of training and averaged over the final 10000 epochs.

Panel of Figure 24 — Figure 24 : Pseudo-2D case, $\zeta=15^{\circ}$ : centreline height profiles at six time instants. PINN (solid blue) and TITAN2D (dashed red).

Panel of Figure 25 — Figure 25 : Pseudo-2D case, $\zeta=15^{\circ}$ : top-view depth contours at six time instants. Contour levels are identical across all panels.

Panel of Figure 26 — Figure 26 : Pseudo-2D case, $\zeta=20^{\circ}$ : centreline height profiles at six time instants. PINN (solid blue) and TITAN2D (dashed red).

Panel of Figure 27 — Figure 27 : Pseudo-2D case, $\zeta=20^{\circ}$ : top-view depth contours at six time instants. Contour levels are identical across all panels.

Panel of Figure 28 — Figure 28 : Pseudo-2D case, $\zeta=30^{\circ}$ : centreline height profiles at six time instants. PINN (solid blue) and TITAN2D (dashed red).

Panel of Figure 29 — Figure 29 : Pseudo-2D case, $\zeta=30^{\circ}$ : top-view depth contours at six time instants. Contour levels are identical across all panels.

Panel of Figure 30 — Figure 30 : Case C1 ( $1\,\text{kg}$ , $10^{\circ}$ ): centreline (left) and cross-slope (right) height profiles at $t=0.00$ , $0.18$ , $0.35$ , and $0.70\,\text{s}$ . PINN (solid blue) and TITAN2D (dashed red).

Panel of Figure 31 — Figure 31 : Integral flow diagnostics for Case C1 ( $1\,\text{kg}$ , $10^{\circ}$ ): $h_{\max}$ and $\bar{V}$ versus time (top row); centroid displacement and $\bar{V}$ versus runout distance (bottom row). TITAN2D (solid red); PINN (dashed blue).

Figure 32 : Time-resolved IoU for Case C1 (

1\,\text{kg}

10^{\circ}

). Time-mean value:

70.9\,\%

Panel of Figure 33 — Figure 33 : Case C2 ( $1\,\text{kg}$ , $15^{\circ}$ ): centreline (left) and cross-slope (right) height profiles at $t=0.00$ , $0.18$ , $0.35$ , and $0.70\,\text{s}$ . PINN (solid blue) and TITAN2D (dashed red).

Panel of Figure 34 — Figure 34 : Integral flow diagnostics for Case C2 ( $1\,\text{kg}$ , $15^{\circ}$ ). Layout as in Figure 31 .

Figure 35 : Time-resolved IoU for Case C2 (

1\,\text{kg}

15^{\circ}

). Time-mean value:

80.7\,\%

Panel of Figure 36 — Figure 36 : Case C3 ( $2.5\,\text{kg}$ , $10^{\circ}$ ): centreline (left) and cross-slope (right) height profiles at $t=0.00$ , $0.24$ , $0.48$ , and $0.95\,\text{s}$ . PINN (solid blue) and TITAN2D (dashed red).

Panel of Figure 37 — Figure 37 : Integral flow diagnostics for Case C3 ( $2.5\,\text{kg}$ , $10^{\circ}$ ). Layout as in Figure 31 .

Figure 38 : Time-resolved IoU for Case C3 (

2.5\,\text{kg}

10^{\circ}

). Time-mean value:

73.1\,\%

Panel of Figure 39 — Figure 39 : Repeatability study for C4 ( $2.5\,\text{kg}$ , $15^{\circ}$ ): centerline (left) and cross-slope (right) height profiles at $t=0.00$ , $0.23$ , $0.47$ , and $0.95\,\text{s}$ . Each solid curve is one of three independent PINN runs. TITAN2D reference (dashed red).

Panel of Figure 40 — Figure 40 : Final deposit profiles for C4 across three independent training runs (solid lines). TITAN2D (dashed red) and Maeno et al. (2013) experimental data (open circles) are shown for reference.

Panel of Figure 41 — Figure 41 : Time-resolved error metrics across three independent training runs for C4. Each curve corresponds to one run; the bandwidth quantifies run-to-run variability.

Profile	Point	$x$ (cm)	$y$ (cm)
Centerline ( $y=0$ )	P1	$-10$	$0$
	P2	$-5$	$0$
	P3	$5$	$0$
	P4	$10$	$0$
	P5	$25$	$0$
Cross-slope ( $x=0$ )	P6	$0$	$-17.0$

Appendix

Table 8 : Coordinates of the ten training data points used for C4. All values are in centimetres.

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Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance

Xiaofeng Liu

Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA, USA · Institute for Computational and Data Sciences, Pennsylvania State University, University Park, PA, USA

Parametric Physics-Informed Neural Network Reynolds-Averaged Navier-Stokes Surrogate Models Anisotropic Loss Landscapes Guidance

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Physics-informed neural networks (PINNs) are a simple surrogate-modelling paradigm for partial differential equations, but their standard strong-form residual formulation is ill suited to the shallow water equations (SWE). It cannot enforce local conservation, handle discontinuities, or leverage the boundary-conforming unstructured meshes used in real-world applications. We introduce ``Data-Guided FVM-PINN'', a framework that replaces the strong-form residual with a differentiable, well-balanced Roe Riemann-solver finite-volume (FVM) loss evaluated on unstructured meshes. The major finding is that physics-only FVM-PINN training often fails on realistic 2D problems: the network collapses to a trivial low-momentum state that nearly satisfies the FVM-PINN residual but bears no resemblance to the true flow. A loss-landscape diagnostic shows that the FVM-PINN loss at zero momentum is only about

7\times

larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a

310\times

separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just

200

random velocity measurements drop the velocity-field

L_2

error by

22\times

versus physics-only;

50

measurements still deliver a

7\times

reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field

L_2

\sim$$23\%

in the sparse-data regime and is essentially neutral when dense reference data is available. On a real-world Savannah River reach (

1306

cells,

3600

~s simulation, five Manning zones), the framework constructs an accurate surrogate from SRH-2D anchor data, with time-window decomposition reducing error monotonically via progressive initial-condition handoff.

Abstract

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