Wildfire Simulation with Differentiable Randers-Finsler Eikonal Solvers

1. Problem Statement

The paper addresses the challenge of inverse problems in wavefront propagation, specifically the need to recover underlying propagation parameters (such as speed and directionality) from observed arrival times.

• Context: Standard Eikonal solvers (like Fast Marching or Fast Sweeping) solve the forward problem: given a speed function, compute arrival times. However, applications like seismic tomography, robotics path planning, and wildfire modeling require the inverse problem: given observed fire spread or wavefront data, infer the environmental factors (terrain, wind, fuel) driving that spread.
• The Bottleneck: Existing solvers are not easily differentiable. They rely on discrete operations (e.g., heap-based priority queues in Fast Marching or discrete stencil selection in Fast Sweeping) that create non-differentiable points, preventing the use of gradient-based optimization (backpropagation) to learn parameters directly from data.
• Specific Application: Wildfire propagation is inherently anisotropic (speed depends on direction due to wind and slope) and asymmetric (fire moves faster downwind/uphill than upwind/downhill). Standard isotropic models fail to capture this, while complex physics-based simulators are often too slow for learning or lack differentiability.

2. Methodology

The authors propose a differentiable framework for the Randers-Finsler Eikonal equation, combining a fast forward solver with an implicit differentiation backward pass.

A. Mathematical Formulation

The framework generalizes the standard Eikonal equation to the Randers-Finsler metric:
$$(\nabla T - b)^\top G^{-1} (\nabla T - b) = 1$$

• $T(x)$: Arrival time field.
• $G(x)$: A symmetric positive-definite metric tensor encoding anisotropic propagation speeds (e.g., terrain slope effects).
• $b(x)$: A drift vector field encoding asymmetric directional bias (e.g., wind direction).
  This formulation unifies isotropic, Riemannian, and asymmetric propagation into a single model.

B. Forward Solver: Differentiable Fast Sweeping

• Algorithm: The authors use a Fast Sweeping Method (FSM) on Cartesian grids. Unlike Fast Marching (which uses a heap), FSM iterates through the grid in fixed directional sweeps (e.g., left-to-right, top-to-bottom).
• Stencil Selection: At each grid point, the solver selects the optimal "triangular stencil" (a pair of upwind neighbors) that minimizes the arrival time. This selection is discrete and non-smooth.
• Optimization: The implementation uses a column/row decomposition to enable GPU parallelization, achieving $O(n)$ complexity per iteration (where $n$ is the number of grid points) and converging in 2–3 iterations regardless of grid size.

C. Backward Pass: Implicit Differentiation

Instead of unrolling the solver iterations (which is memory-prohibitive), the authors use Implicit Differentiation:

1. Fixed Point: The converged solution $T$ satisfies a residual equation $R(T, \theta) = 0$, where $\theta$ represents the parameters ($G$ and $b$).
2. Adjoint System: By applying the Implicit Function Theorem, the gradient $\frac{dL}{d\theta}$ is computed by solving a sparse, lower-triangular adjoint system:
   $$\left(\frac{\partial R}{\partial T}\right)^\top \lambda = \left(\frac{\partial L}{\partial T}\right)^\top$$
3. Efficiency: Because the Jacobian is sparse and triangular (due to the causality of the forward solve), the adjoint system is solved via a single reverse-time back-substitution pass. This avoids storing intermediate states and reduces memory usage significantly.

D. Handling Non-Smoothness

A key theoretical contribution is the analysis of stencil boundaries (where the solver switches from one stencil to another).

• The authors prove that these boundaries form a measure-zero set (1D curves in 2D space).
• They show that while the solution is non-differentiable at these boundaries, the gradients are stable under small perturbations. The method effectively differentiates through the "winning" stencil, yielding a valid element of the Clarke subdifferential, which is sufficient for stochastic gradient descent convergence.

E. Learning Architecture

For wildfire applications, a neural network encoder maps environmental covariates (elevation, vegetation, wind, humidity) to the Randers parameters ($G, b$). The system is trained end-to-end by minimizing the difference between predicted and observed fire arrival times.

3. Key Contributions

1. Differentiable Randers-Finsler Solver: The first framework to combine Fast Sweeping with implicit differentiation for anisotropic and asymmetric metrics, enabling gradient-based learning of propagation parameters.
2. Theoretical Stability Analysis: A rigorous proof that stencil selection non-differentiability is a measure-zero issue and that gradients remain stable under perturbation, validating the use of standard optimizers.
3. Computational Efficiency:
   • Forward: Up to 1400× speedup over Jacobi iteration.
   • Backward: Up to 755× speedup over finite difference approximations for gradient computation.
4. Real-World Application (Wildfire): Successful application to the Sim2Real-Fire dataset, demonstrating the ability to learn complex, spatially-varying fire spread dynamics from observational data.

4. Results

Synthetic Validation

• Forward Accuracy: Achieved <1% relative error across isotropic, anisotropic, and combined Randers-Finsler configurations.
• Gradient Verification: Implicit gradients matched finite differences exactly (error < $10^{-6}$) at interior points.
• Inverse Recovery:
  • Recovered isotropic metrics with ~5.6% error.
  • Recovered drift fields with <3% error.
  • Multi-source learning (using multiple ignition points) reduced errors by 45% compared to single-source, as it provides complementary geometric constraints.

Wildfire Simulation (Sim2Real-Fire)

• Within-Scene Generalization: Models trained on specific landscapes achieved a mean test correlation of 0.824 for predicting fire arrival times on unseen fires within the same scene.
• Cross-Scene Transfer: A model trained on 11 scenes generalized to 4 unseen geographic regions with a correlation of 0.766, proving the model learns universal physics rather than memorizing specific locations.
• Model Capacity: A convolutional encoder (116K parameters) outperformed per-pixel optimization (up to 1.7M parameters) on small fires, demonstrating that the primary challenge is learning generalizable covariate-to-metric mappings, not model expressiveness.
• Real-World Transfer: A model trained purely on synthetic simulation data transferred zero-shot to real satellite wildfire data, achieving a correlation of 0.588, outperforming a model trained directly on sparse real data.

5. Significance

This work bridges the gap between classical numerical PDE solvers and modern deep learning.

• Scientific Impact: It provides a rigorous, differentiable primitive for learning physical propagation laws (anisotropy and drift) directly from data, applicable to seismology, medical imaging, and robotics.
• Practical Impact: For wildfire management, it offers a scalable, data-driven alternative to computationally expensive physics simulators. It enables the creation of "digital twins" for fire prediction that can be trained on vast amounts of simulation data and adapted to real-world scenarios with sparse observations.
• Methodological Impact: It demonstrates that discrete numerical methods (like Fast Sweeping) can be made differentiable and efficient for end-to-end learning, challenging the notion that only smooth, continuous solvers (like PINNs) are suitable for gradient-based optimization.