Procedural Thermal Forest Simulation

• Procedural Thermal Forest Simulation is a computational approach that generates forest landscapes by integrating procedural vegetation geometry with thermal processes using state-space models and PDEs.
• It applies patch-based decomposition and multi-directional scanning to simulate heat transfer, combustion dynamics, and spatial variability efficiently with linear time complexity.
• The method supports applications in wildfire risk analysis, visual effects, and environmental modeling, offering scalable performance with quantified trade-offs against transformer-based approaches.

Procedural Thermal Forest Simulation refers to the algorithmic generation and evolution of forest scenes, with an explicit focus on simulating thermodynamic and structural processes affecting vegetation—typically under the presence of heat transfer, combustion dynamics, or related energy-driven ecological effects. The field draws upon procedural modeling, physics-based simulation, state-space modeling, and high-performance computing, and is relevant in visual effects, environmental modeling, remote sensing, and wildfire risk analysis.

1. Mathematical and Algorithmic Foundations

Procedural forest simulation commonly begins with a generative model for vegetation geometry (procedural foliage, branching, undergrowth) and overlays thermal processes modeled by coupled partial differential equations (PDEs), cellular automata, or discrete state-space models. The continuous-time state-space model (SSM), as formalized in numerous visual SSM works, provides a foundation:

• Continuous-time LTI systems: $h'(t) = A h(t) + B x(t),\ y(t) = C h(t) + D x(t)$, with $h$ the latent state (e.g., local energy content, fuel type, moisture), $x$ external input (heat sources, wind), and $y$ observables (temperature, combustion state).
• Zero-Order Hold Discretization: For computational tractability, time discretization is used: $h_t = \bar{A} h_{t-1} + \bar{B} x_t$, $y_t = C h_t + D x_t$, where $\bar{A} = \exp(\Delta A)$, $$\bar{B} = (\Delta A)^{-1} (\exp(\Delta A) - I) \Delta B$$ (Wu et al., 6 May 2024).

For spatially extended forests, these dynamics may be applied patch-wise or across a grid, with each patch representing a canopy voxel or ground subregion. The system matrices $(A,B,C,D)$ can encode localized heat transfer, radiative exchange, or combustion kinetics, with input conditionings to encode wind, humidity, or topological variation.

Bidirectional and multi-directional scanning in SSM architectures (left–right, top–bottom, diagonal) facilitate efficient propagation of thermal information across the forest structure while maintaining linear time complexity (Wu et al., 6 May 2024). For detailed fire spread, coupling with frequency-domain solvers or FFT-based spectral representations is established for modeling the propagation of thermal fronts (Zhang et al., 29 May 2024).

2. Representing Forest Geometry and Dynamics

Procedural forests leverage stochastic L-systems, rule-based growth simulators, or recursive partitioning to generate vegetation structure. For thermal simulation, these generate spatial indices used to parametrize local SSMs or PDE solvers. The feature tensor $X \in \mathbb{R}^{C \times H \times W}$ may hold geometric, biophysical, and initial thermodynamic states.

• Patch-Based Decomposition: Each generated patch or segment is mapped to a set of state variables—energy content, combustibility, and local moisture. Modern SSM frameworks vectorize each patch in multiple scan orders (row/column, diagonal, and reversals) to ensure robust spatial correlation modeling (Wu et al., 6 May 2024).
• Heat and Combustion Dynamics: The underlying state update simulates energy injection (solar, lightning, anthropogenic), advective transfer (wind-driven fire spread), and radiative/conductive losses. Parameterization of $A$ (heat loss/gain rates), $B$ (external sources), and $C$ (measurement extraction) is adjusted for local vegetation type and density.
• Thermodynamic Boundary Conditions: Procedural simulation enables automated imposition of boundary and initial conditions (terrain, weather, human intervention scenarios) at image- or patch-level, with state evolution responding accordingly.

3. State-Space Model Implementation and Complexity

Visual SSM-based simulation excels at balancing global receptive field and computational scalability. Utilizing the VSSM-CA approach (Wu et al., 6 May 2024), core architectural steps include:

• Feature Extraction: Input tensors $Z$ are normalized and processed via FC/conv/SiLU activations, then SSM scan is performed in each patch using the ‘V-S6’ selective scan across four spatial orders (Eq. (7)-(11)). The update $h_t = \bar{A} h_{t-1} + \bar{B} x_t$ runs per vector with $O(DN)$ FLOPs (D = patch area, N = latent state).
• Gating and Fusion: Each SSM output stream is modulated with a learned gating map, fusing heat signals across patches and spatial routes, simulating spatial interdependence (e.g., fire jumping, embers transported by wind).
• Complexity: The system attains linear computational complexity in number of spatial positions and state size, with overall per-block FLOPs scaling as $O(N D)$ per scan direction, or $O(4 N D)$ for four-way scanning. All-SSM updates preserve this efficiency and permit large-scale simulation (Wu et al., 6 May 2024), outperforming quadratic-complexity transformer alternatives for large terrain grids.

4. Channel State Information and Adaptation

For physically realistic simulations, variable channel conditions—such as spatial heterogeneity in fuel, moisture, or local wind—must adaptively influence thermodynamic propagation. The VSSM-CA channel adaptation block enables incorporation of such context (Wu et al., 6 May 2024):

• Context Embedding: Channel state information (CSI)—e.g., wind vectors, local humidity, or firefighting intervention—are encoded via sinusoidal position encoding, FC layers, and nonlinearity, yielding an adaptation vector $u \in \mathbb{R}^m$ (typically $m = 128$).
• Patch Injection: Each VSSM-CA block injects the embedded CSI into patchwise latent states, ensuring local dynamics (combustion thresholds, heat capacity) reflect environment or suppression state.
• Global-Local Consistency: All blocks in the encoder (and separately, the decoder for inverse tasks) share the same context vector, allowing the backbone model to maintain consistency across simulation depths and spatial resolutions.

This approach aligns with adaptive channel coding in wireless JSCC, but the procedural forest context enables spatially resolved, physically coherent adaptation to landscape features or dynamic weather fronts.

5. Quantitative Performance and Trade-Offs

Empirical studies document major efficiency gains for such state-space approaches:

• Parameter and Computational Efficiency: With matched model depth, VSSM-CA achieves $0.48$ dB PSNR gain over Swin Transformer baselines in simulated image transmission while using $53.3\%$ of MACs, $53.8\%$ of parameters, and $44.9\%$ of inference delay (Wu et al., 6 May 2024).
• Scalability: Increasing VSSM-CA depth linearly grows compute and parameters, with monotonic gains in reconstruction or tracking fidelity. Trade-off curves demonstrate that state-space models maintain superior accuracy-to-cost ratio compared to transformer backbones, critical for large simulation grids encountered in ecosystem modeling.
• Sensitivity to Channel Adaptation: Under varying adaptation regimes (e.g., altered wind scenario), VSSM-CA maintains $+0.25$–$0.65$ dB PSNR advantage over transformer-based methods, emphasizing robustness in dynamically evolving physical settings (Wu et al., 6 May 2024).

6. Extensions and Integration in Physical Simulation Pipelines

Integration of procedural thermal forest simulation within broader physical or environmental modeling pipelines is an active area:

• Multi-Scale and Hierarchical Modeling: State-space models can be composed hierarchically—fine-scale SSMs handle foliage and underbrush, while coarser blocks model canopy and terrain-scale effects, enabling efficient multiscale wildfire progression simulation.
• Spectral and Frequency-Domain Enhancement: For high-resolution or oscillatory thermodynamics (e.g., radiative feedback, fire whirls), augmenting spatial scans with FFT-based or attention-like spectral blocks improves expressivity and accuracy (Zhang et al., 29 May 2024).
• Cross-Domain Applications: Beyond wildland fire simulation, similar procedural SSM pipelines are applicable to urban heat islands, remote sensing of thermal anomalies, or interactive visualization in computer graphics.

7. Limitations and Future Directions

Procedural thermal forest simulation via SSMs remains an active frontier:

• Physical Fidelity: While linear state-space dynamics support tractable simulation at scale, nonlinear combustion or moisture transport phenomena may require integrating neural or physics-informed extensions, as well as explicit PDE coupling.
• Heterogeneous Adaptation: Scalably capturing highly variable forest structures or stratified environments demands further development of adaptive gating, multi-branch SSMs, or deformable scan strategies.
• Data-Driven Calibration: Empirical parameterization or end-to-end learning from aerial/remote sensing fire progression data can further refine procedural models, but requires large, labelled datasets and domain adaptation workflows.

Ongoing work continues to refine the interplay between computational efficiency, spatial expressivity, and physical fidelity in procedural thermal forest simulation frameworks based on visual state-space modeling (Wu et al., 6 May 2024).