1 Introduction↩︎

Wildfires in the United States consume millions of acres each year, posing significant threats to ecosystems, human life, and property[1]. To mitigate these risks, fire managers employ prescribed burns — intentional, controlled fires in designated forest areas designed to reduce fuel loads that can cause dangerous wildfires, enhance forest health, and help manage pests[2]. However, these burns are only initiated under specific conditions to ensure safety and prevent unintended consequences [3]. Fire managers rely on process-based models such as FARSITE [4] and QUIC-Fire [5] to predict the behavior and trajectory of prescribed fires under varying weather conditions (primarily wind speed and wind direction), fuel density in the area, and pre-defined ignition conditions. These process-based models simulate physical fire dynamics to assess burn risks, but their computational demands often prevent real-time decision-making [6], [7].

While purely data-driven deep learning architectures, such as CNNs [8], [9] and U-Nets [10], offer computational speedup, they are ill-suited for complex multi-point ignitions and frequently predict non-physical behaviors like spontaneous fuel regeneration. To counteract this, existing physics-guided ML approaches attempt to constrain outputs using partial differential equations (PDEs) [6], [11]. However, these rigid mathematical formulations are often highly unstable, fail to converge, or suffer from well-documented gradient flow pathologies [12], [13]. Our proposed surrogate framework addresses these gaps by embedding domain-specific physical constraints as "soft" penalties, providing the guidance of physics without the numerical fragility of strict equation solving.

In this paper, we present a framework to incorporate physics components into deep learning models that can work across different architectures. Specifically, we present a comparative analysis with and without physics-guided loss for three deep learning models, namely Convolutional LSTM (ConvLSTM) [14], Adaptive Fourier Neural Operator (AFNONet) [15], and Video Vision Transformer (ViViT) [16], for the task of spatiotemporal fuel density change.

2 Dataset and Problem Formulation↩︎

We utilize an ensemble of QUIC-Fire simulations of fire spread, measuring fuel density (mass of fire fuel per unit volume) over time in an area. Each simulation runs for 50 seconds on a 300 \(\times\) 300 spatial grid with flat grassland. The dataset comprises all possible combinations of 7 wind speed values (ranging from 1 m/s to 15 m/s), 11 wind direction values (from 230 to 330 degrees), 4 distinct ignition pattern types (aerial, inward, outward, strip north). The wind speed and wind direction remain constant across the entire spatial grid and all timesteps. The data contains fuel densities in the range [\(0,0.7kg/m^2\)]. To ensure effective learning, preprocessing steps include normalization of wind speed and wind direction using min-max scaling, and incorporation of a source map. The source map is a 2d array that represents fuel density evolution over time at the lowest wind speed (1 m/s), and the same wind direction and ignition pattern as the input data, allowing the model to learn fuel density propagation patterns. The final dataset consists of four feature channels: source map, wind direction, wind speed, and ignition pattern.

Our objective is to model the evolution of fuel density as a prescribed fire propagates under different environmental conditions. Specifically, we aim to develop a data-driven emulator of QUIC-Fire, using simulated runs as training data. Each training example, indexed by \(i\), is represented by an input tensor \(X_i \in R^{T \times H \times W \times C}\) where T denotes the number of timesteps in the sequence, H and W define the spatial grid dimensions, and C represents the number of input channels. The corresponding output tensor for fuel density predictions is given by \(Y_i \in R^{T \times H \times W \times 1}\). In our setup, wind speed and wind direction remain static across time. Ignition patterns and source fuel density maps, on the other hand, evolve dynamically over time. By structuring the dataset in this manner, our approach allows the model to learn complex spatiotemporal interactions between fire behavior and environmental conditions, enabling improved predictions of fuel density evolution in prescribed fire scenarios.

3 Methods↩︎

We evaluated three deep learning architectures to represent distinct spatiotemporal paradigms: ConvLSTM uses a convolutional neural network (CNN) for spatial structure and a long short-term memory (LSTM) network for temporal dynamics; ViViT leverages self-attention to model long-range temporal dependencies; and AFNONet utilizes Fourier domain mixing to efficiently capture global dependencies.

3.1 ConvLSTM↩︎

Our ConvLSTM model adapts the architecture proposed by Chatterjee et al. [2]. While the references work utilizes a four-layer stacked ConvLSTM with Batch Normalization, we simplify this design to a two-layer structure (64 filters, \(3\times3\) kernels) to prevent overfitting on limited simulation data.

We introduce two critical architectural enhancements to the standard ConvLSTM to support fuel density modeling:

1. Residual Connections: We attach an auxiliary 3D convolutional head to the first LSTM layer to allow injection of physics-based gradients directly into the intermediate layers via residual connections [17], preventing gradient vanishing during backpropagation through time.

2. Spatiotemporal Smoothing: We use a 3D kernel of (\(3\times3\times3\)) which integrates the information across adjacent time steps and pixels, ensuring temporal continuity in the predicted fire front.

To maintain numerical stability, we use LeakyReLU (\(\alpha=0.1\)) to prevent the "dead neuron" collapse often observed in sparse data like fire maps.

3.2 Video Vision Transformers (ViViT)↩︎

We utilize ViViT [16], a transformer-based architecture originally designed for video classification, to model the spatiotemporal evolution of fuel density. ViViT processes its input by tokenizing the video sequence using tubelet embeddings, which serve as the spatiotemporal representation of the fuel grid in our use case.

To reconstruct the fuel density map, we added additional processing layers to our ViViT. Token embeddings are first mapped back to their spatial regions through a token-to-tubelet reconstruction process: A Multi-Layer Perceptron (MLP) projects each token into its respective tubelet dimensions. The reconstructed tubelets are then reshaped into the predicted tensor, followed by a Conv3D layer, which refines the temporal dependencies, ultimately generating the final fuel density prediction for the next timestep. This structured approach allows ViViT to effectively model the spatiotemporal dynamics of fire spread, leveraging transformer-based self-attention for accurate fuel density forecasting.

3.3 Adaptive Fourier Neural Operator Network (AFNONet)↩︎

The Adaptive Fourier Neural Operator Network (AFNONet) [15], also a transformer-based deep learning model, replaces the self-attention mechanism, which has complexity of \(O(N^2)\), with Fourier-domain token mixing, reducing complexity to \(O(N \log N)\) while effectively capturing the fine-scale differential physics of fluid dynamics [18].

We adapt the standard AFNO architecture for spatiotemporal forecasting by implementing an AFNONet_Seq2Seq model. While standard AFNO models typically function as autoregressive operators mapping \(X_t \to X_{t+1}\), our adaptation leverages 3D tokenization to map a context sequence \(X_{in}\) directly to a prediction sequence \(Y_{out}\) in a single forward pass. The input tensor \(X \in \mathbb{R}^{B \times C \times T_{in} \times H \times W}\) is processed by a 3D Patch Embedding layer. The volume is partitioned into non-overlapping "tubelets" of size \((time, height, width)\). A 3D convolution projects each tubelet into a latent embedding dimension \(D\), effectively compressing local spatiotemporal dynamics into a single token grid. The token grid is reshaped to merge temporal and vertical spatial dimensions, allowing the AFNO blocks to process the sequence as a unified field.

Unlike standard classification heads, our decoding head performs a high-dimensional expansion to reconstruct the full video volume. A linear projection maps each latent token back to the pixel space of its original tubelet: \(\hat{v} \in \mathbb{R}^{C_{out} \times p_t \times p_h \times p_w}\). Finally, a rearrangement operation (un-patching) merges these local tubelet predictions back into the global grid structure, yielding the continuous output sequence \(Y_{out} \in \mathbb{R}^{B \times T_{out} \times C_{out} \times H \times W}\).

4 Integrating Physics Constraints↩︎

To enforce physical consistency, we define a composite loss function composed of the following loss components:

1. Base MSE (\(\mathcal{L}_{\text{MSE}}\)): We employ standard pixel-wise Mean Squared Error(MSE) as an objective to minimize the discrepancy between the ground truth and prediction.

2. Fuel Transport (\(\mathcal{L}_{\text{fuel}}\)): Since fuel consumption is irreversible, we penalize non-physical fuel regeneration (i.e., positive temporal gradients) by taking MSE of pixels with increased fuel density over time. \[\mathcal{L}_{\text{fuel}} = \frac{1}{N} \sum_t \mathbb{1}_{(\hat{F}_t > \hat{F}_{t-1})} \odot \left\| \hat{F}_t - F^{GT}_t \right\|^2,\] where \(N\) is the number of timesteps, \(\hat{F}_t\) is the 2d fuel density prediction map at time \(t\), and \(F^{GT}_t\) is the 2d fuel density map at \(t\) obtained from the ground truth.

3. State-Weighted Losses (\(\mathcal{L}_{\text{burn}}, \mathcal{L}_{\text{unburn}}\)): To weight burned and unburned pixels independently, we separated loss components based on a differentiable mask calculated through the temperature-scaled sigmoid function at time \(t\) as: \[\label{eq:diffmask} P^{GT}_t=\sigma\left({\left({F^*} - F^{GT}_t\right)/\mathcal{T}}\right),\tag{1}\] where \(F^*=0.665\) is the heuristically determined fuel density threshold from the ground truth fuel densities, such that \(F^{GT}_t > F^*\) are the unburned pixels, and \(F^{GT}_t\leq F^*\) are the burned pixels. We scale the sigmoid function by \(\mathcal{T}=0.02\), narrowing the intermediate values between \(0\) and \(1\) to resemble a differentiable mask that can backpropagate during training. Utilizing \(P^{GT}\), we calculate the burned and unburned loss components through \[\begin{align} \label{eq:statelosses} \begin{aligned} \mathcal{L_\text{burn}}&=\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}P^{GT}_t\odot L_{MSE}\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}\\ \mathcal{L_\text{unburn}}&=\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}(\mathbb{1}-P^{GT}_t)\odot L_{MSE}\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}} \end{aligned} \end{align}\tag{2}\] This formulation ensures that \(\mathcal{L}_{\text{burn}}\) dominates in active fire regions, while \(\mathcal{L}_{\text{unburn}}\) stabilizes the background.

4. Rate of Spread (\(\mathcal{L}_{\text{ROS}}\)): For a fire propagating with eastward wind, the rate of spread (ROS) is the speed at which the leading edge in the \(x-\)axis of a fire advances over the landscape. We can calculate the leading edge of an eastward-propagating fire at time \(t\) through \[\label{eq:argmax} \nu(t)=\Delta x\cdot \verb|argmax|_j(F_{ij}(t)),\tag{3}\] where \(\Delta x=2m\) is the grid size and \(\verb|argmax|_j\left(F_{ij}(t)\right)\) is the index of the pixel with maximum fuel density at time \(t\). Using the Eq. 3 , we can calculate the ROS through \[\label{eq:ros} \text{ROS}(t) = \frac{1}{t}\left(\nu(t)-\nu(0)\right).\tag{4}\] However, the argmax function in Eq. 3 is nondifferentiable, which prevents backpropagation during training. Therefore, we approximate the leading edge using a series of differentiable equations as follows.

We first project the 2D differentiable mask in Eq. 1 to a 1D transverse profile through \(p_j(t) = \max_i P_t^{GT}\) and calculate a location score, \(S_j(t) = j + \lambda (p_j(t) - 1)\), to suppress noise from non-burning columns. Through the use of temperature-scaled softmax of the location scores, \(\verb|softmax|(S_j(t)/\mathcal{T})\), we calculate an approximate leading edge for the ground truth fuel map at time \(t\), \(\nu^{GT}(t)=\sum_j j\cdot p_j(t) \cdot f_j(t)\). We then calculate the approximate ground-truth rate of spread, \(\text{ROS}^{GT}(t)\), through the approximate leading edge using Eq. 4 . For the predicted approximate rate of spread, \(\widehat{\text{ROS}}(t)\), we use the predicted fuel density map, \(\hat{F}_t\), instead of the ground truth, \(F_t^{GT}\), while calculating the differentiable mask in Eq. 1 . We constrain the kinematic velocity of the fire front by minimizing the error between the predicted differentiable rate-of-spread (ROS) and the ground truth. ROS is calculated through \[\mathcal{L}_{\text{ROS}} = \left\| \widehat{\text{ROS}}(t) - \text{ROS}^{GT}(t) \right\|^2.\]

5. Total Loss (\(\mathcal{L}_\text{total}\)): To enforce physical consistency, we propose WiFireLoss, a composite objective function that aggregates fuel conservation, state consistency, and kinematic elements. The total loss is given by: \[\begin{align} \begin{aligned} \mathcal{L}_{\text{total}} &= \lambda_{\text{MSE}} \mathcal{L}_{\text{MSE}} + \lambda_{\text{fuel}} \mathcal{L}_{\text{fuel}} \\ &+ \lambda_\text{burn} \mathcal{L}_{\text{burn}} + \lambda_\text{unburn} \mathcal{L}_{\text{unburn}} \\ &+ \lambda_\text{ROS} \mathcal{L}_{\text{ROS}}, \end{aligned} \end{align}\] where \(\lambda_{MSE}\), \(\lambda_{fuel}\), \(\lambda_{burn}\), \(\lambda_{unburn}\), and \(\lambda_{ROS}\) are the weights associated with the corresponding loss components.

5 Experimental Setup and Results↩︎

5.1 Experimental Setup↩︎

All models were implemented in PyTorch using bfloat16 mixed precision on a single NVIDIA RTX 3090 Ti. Training utilized the AdamW optimizer with a cosine annealing schedule (Peak \(LR=10^{-3} \to 10^{-5}\)). Models are trained using a context length of 6 frames (\(T_{in}=6\)) to predict a 3-frame future window (\(T_{out}=3\)). For example, given an input sequence of timesteps \(t_1, ..., t_6\), the model predicts \(t_7, t_8, t_9\). Input and output images are \(296\times296\) pixels. The temporal stride is 1, so the next input sequence is \(t_2,...,t_7\).

• ConvLSTM: Trained over 15 epochs with a batch size of 4 and an early stopping patience of 5 epochs. The hidden dimension size is 64.

• AFNONet: Configured with a 3D patch size of \(8\times8\times3\;(H\times W\times T)\), an embedding dimension of 256, and 4 layers with 4 spatial frequency blocks. Trained for 50 epochs with a batch size of 8 and patience of 10 epochs.

• ViViT: Utilizes 3D tubelet tokens of shape \((t=3,h=4,w=4)\). The backbone consists of 4 layers, 4 attention heads, an embedding dimension of 256, and an MLP hidden dimension of 512.

We use the following loss component weights (\(\lambda\)): \(\lambda_{MSE}=1\), \(\lambda_{fuel}=0.01\), \(\lambda_{ROS}=0.01\), \(\lambda_{burn}=0.1\), \(\lambda_{unburn}=0.1\). These \(\lambda\) values were chosen to capture the relative contribution of each component (e.g., burned/unburned are less important than MSE, but more important than fuel transport and ROS). After evaluating several sets of values, these provided the best balance between convergence and training stability.

5.2 Experimental Results↩︎

Table 1 presents the performance evaluation across different metrics, including base MSE, state-based errors (burned/unburned), fuel transport violation, and ROS error. The results show that integrating the physics-guided WiFireLoss improves fuel density prediction performance across the tested architectures. This demonstrates the generality of the approach for physics-guided machine learning and its applicability to different deep learning models. Importantly, the addition of physics loss terms reduces error means as well as standard deviations, indicating improved and more stable prediction performance.

Fig 2 illustrates the results of training with WiFireLoss. Fuel density prediction at timestep 42 for all three models are displayed, along with the ground truth. These images show that all models are able to predict fuel density with high fidelity.

6 Conclusion↩︎

This work develops physics-guided spatiotemporal ML approaches to predict and emulate fuel density evolution over time under different wind conditions and varying ignition patterns. The objective of this project is to enable fire managers to adapt their plans to achieve a burn’s intended objective with changing environmental conditions. Our approach uses a loss function with physical constraints to reduce physical inconsistencies in predictions, including fuel transport discrepancies, over-burning in unburned areas, and over-estimating fuel in burned areas. Our framework can be extended to other spatiotemporal models for learning physically consistent fuel densities. We demonstrate that integrating physics-based constraints into ML models improves fuel density prediction for prescribed fire management.

Future work includes evaluating the proposed method on extended-length fire sequences and on diverse fuel types and topography. We also plan to investigate incorporating other features and additional physical constraints.

Acknowledgment↩︎

The authors would like to thank the Societal Computing and Innovation Lab (SCIL) at the San Diego Supercomputer Center, University of California San Diego for their support of this study. This work was funded in part by San Diego Gas and Electric (SDG&E) and NSF Award 2134904.

References↩︎