Exploring Physics-Informed Neural Networks for Crop Yield Loss Forecasting (2501.00502v1)

Abstract: In response to climate change, assessing crop productivity under extreme weather conditions is essential to enhance food security. Crop simulation models, which align with physical processes, offer explainability but often perform poorly. Conversely, ML models for crop modeling are powerful and scalable yet operate as black boxes and lack adherence to crop growths physical principles. To bridge this gap, we propose a novel method that combines the strengths of both approaches by estimating the water use and the crop sensitivity to water scarcity at the pixel level. This approach enables yield loss estimation grounded in physical principles by sequentially solving the equation for crop yield response to water scarcity, using an enhanced loss function. Leveraging Sentinel-2 satellite imagery, climate data, simulated water use data, and pixel-level yield data, our model demonstrates high accuracy, achieving an R2 of up to 0.77, matching or surpassing state-of-the-art models like RNNs and Transformers. Additionally, it provides interpretable and physical consistent outputs, supporting industry, policymakers, and farmers in adapting to extreme weather conditions.

• The paper proposes a novel physics-informed RNN that integrates machine learning with FAO-based physical models to forecast crop yield loss under water scarcity.
• It uses an LSTM backbone and a custom loss function to sequentially estimate biophysical properties (ET_a and K_y) while enforcing physical consistency.
• Experiments show the PI-RNN achieves an R² of 0.77 using high-resolution Sentinel-2 and weather data, outperforming standard RNNs and simulation models.

This paper, "Exploring Physics-Informed Neural Networks for Crop Yield Loss Forecasting" (2501.00502), addresses the critical need to accurately forecast crop yield loss under extreme weather conditions, particularly water scarcity, in the context of climate change.

Problem: Traditional crop simulation models, while explainable and grounded in physical principles, often struggle with high-resolution data, are computationally expensive, require extensive calibration, and can be inaccurate due to simplified assumptions. Pure data-driven ML models excel at scalability and accuracy with complex data but act as black boxes, lacking interpretability and often failing to adhere to underlying physical principles of crop growth. This gap limits their utility for understanding crop physiology and supporting informed decision-making. Specifically, assessing yield loss due to water scarcity is challenging because accurately determining actual evapotranspiration ($ET_a$) at a high resolution is difficult, making the classic FAO yield response to water equation hard to apply precisely in practice.

Proposed Solution: The authors propose a novel physics-informed neural network (PI-RNN) approach that combines the data-handling capabilities of ML with the interpretability of physical models. The core idea is to use an RNN to learn key biophysical properties, actual water use ($ET_a$) and crop sensitivity to water scarcity ($K_y$), at the pixel level. These learned properties are then used to estimate yield loss based on the established FAO equation for crop yield response to water scarcity. Physical consistency is enforced through a specialized loss function.

Methodology:

1. Physical Foundation: The model is built upon the FAO equation for relative yield loss ($Y_l$):

   $$Y_l = \left( 1 - \frac{Y_a}{Y_x} \right) = K_y \left( 1 - \frac{ET_a}{ET_x} \right)$$

   where $Y_a$ is actual yield, $Y_x$ is maximum (potential) yield, $ET_a$ is actual evapotranspiration, $ET_x$ is maximum evapotranspiration, and $K_y$ is the crop yield response factor. The paper assumes $ET_x$ can be accurately simulated using methods like FAO paper-56, focusing on learning $ET_a$ and $K_y$.

2. Neural Network Architecture: A Recurrent Neural Network (RNN), specifically an LSTM backbone, is used to process time series data. The output layer of the RNN is designed to predict the two biophysical properties, $ET_a$ and $K_y$, for each time step. The architecture includes sequential layers with linear transformations, batch normalization, and dropout before the final output layers for $K_y$ and $ET_a$. The RNN is trained to predict these properties sequentially over the growing season.
3. Physics Integration through Loss Function: A two-component loss function enforces physical consistency:

   $$\mathcal{L}_{total} = \lambda_1 \mathcal{L}_{l} + \lambda_2 \mathcal{L}_{phys}$$

* $\mathcal{L}_{l}$: Standard mean squared error (MSE) between the predicted yield loss ($\hat{y_a}$) derived from the FAO equation (using the predicted $ET_a$ and $K_y$) and the ground truth actual yield loss ($y_a$). $\mathcal{L}_{l} = \mathbb{E}[(\hat{y_a} - y_a)^2]$. This term drives the model to accurately predict the final yield loss. * $\mathcal{L}_{phys}$: A penalty term designed to ensure $ET_a$ predictions are physically plausible and contribute meaningfully to solving the FAO equation. This term specifically encourages $ET_a$ values to be within the bounds [0, $ET_x$] and close to $ET_x$ (under non-limiting conditions).

$$\mathcal{L}_{phys} = \mathbb{E} \left[ 1_{\{ET_a < 0\} \cdot (ET_a)^2} + 1_{\{ET_a > ET_x\} \cdot (ET_a - ET_x)^2} + 1_{\{0 \leq ET_a \leq ET_x\} \cdot (ET_a - ET_x)^2} \right]$$

The indicator function $1\{\cdot\}$ applies penalties if $ET_a$ is outside the [0, $ET_x$] range and an MSE-like term within bounds. The $\lambda_1$ and $\lambda_2$ are hyperparameters weighting the two loss components.

1. Data:
   • Ground Truth: Pixel-level combine harvester yield data for cereal crops in Switzerland (2017-2021), with 54,098 samples from 54 yield maps (10m resolution). Yield loss is calculated relative to the maximum yield sample within the dataset ($Y_x$).
   • Simulated Data: $ET_x$ values simulated over time using the FAO paper-56 method.
   • Input Data:
     • Sentinel-2 (S2) L2A multispectral time series (10 bands, 10m resolution) from seeding to harvest.
     • Weather data (total precipitation, temperature) derived from ERA5 (aggregated between S2 time steps).
   • Data modalities (S2 and weather) are combined using early fusion of raw time series.

Experiments and Results:

• Comparison with SOTA ML: The PI-RNN was compared against standard RNN (LSTM) and Transformer models for yield loss prediction using S2 + Weather data.
  • The PI-RNN achieved an $R^2$ of 0.77, outperforming the standard RNN ($R^2=0.75$) and performing similarly to the Transformer ($R^2=0.78$). MAE and RMSE were also comparable or better than the standard RNN.
  • Importantly, unlike the standard ML models, the PI-RNN provides interpretable outputs ($ET_a$, $K_y$).
• Comparison with Simulation Model: The simulation model (predicting using weather data) showed significantly poorer performance ($R^2=-6.92$) compared to the ML models, highlighting the difficulty of applying coarse-resolution simulations at a fine pixel scale and accurately estimating $ET_a$ without detailed field-level data.
• Modality Ablation: Experiments with the PI-RNN using different input modalities showed the importance of S2 data:
  • Weather only: $R^2=0.46$
  • Sentinel-2 only: $R^2=0.75$
  • Sentinel-2 + Weather: $R^2=0.77$
  • This indicates that high-resolution S2 imagery is crucial for capturing spatial variability and achieving high accuracy at the pixel level.
• Interpretable Outputs: The sequential estimations of $ET_a$ and $K_y$ demonstrated physically consistent trends over the growing season. Estimated $ET_a$ was consistently below simulated $ET_x$, reflecting water limitation and associated yield loss. $K_y$ values were generally below 1, suggesting some resilience to water scarcity in the dataset used. Predictions became more accurate closer to harvest.

Implementation Considerations:

• Data Preprocessing: Requires handling time series data from S2 (potential missing values due to clouds need interpolation/imputation) and aligning it with weather data and pixel-level yield observations. Georeferencing and resampling might be necessary.
• Model Architecture: An LSTM is used, which is suitable for sequence data. Implementing the custom two-component loss function is crucial for enforcing physical constraints. Libraries like PyTorch or TensorFlow allow for custom loss functions.
• Hyperparameter Tuning: The weights $\lambda_1$ and $\lambda_2$ in the loss function need tuning. Standard training procedures for RNNs apply, including optimization algorithms (e.g., Adam) and learning rate scheduling.
• Computational Resources: Training RNNs on time series data from high-resolution imagery can be computationally intensive, requiring GPUs. The dataset size (54,098 pixel samples) is significant.
• Scalability: The pixel-level approach can be scaled spatially if sufficient high-resolution data (S2, weather, and ground truth yield) is available for larger areas. Simulating $ET_x$ also needs to be scalable.
• Data Availability: Access to high-quality, georeferenced pixel-level yield data is often a limiting factor for training and validating such models. S2 and ERA5 data are publicly available, but aligning them precisely with yield data requires careful processing.

Practical Applications:

• Yield Loss Assessment: Provides estimates of yield loss at a fine spatial resolution, allowing for targeted interventions or risk assessment.
• Understanding Water Stress: The intermediate outputs ($ET_a$, $K_y$) offer insights into how water availability affects crop performance throughout the season, similar to process-based models but derived from observed data.
• Supporting Decision Making: Provides actionable information for farmers (irrigation planning), policymakers (drought impact assessment, resource allocation), and industry (insurance, supply chain forecasting).
• Early Forecasting: The model can provide increasingly accurate yield loss forecasts as the season progresses, as shown in the results.

Limitations and Future Work (as mentioned in the paper):

• Further validation on larger and more diverse datasets is needed to confirm the model's validity across various crops, regions, and yield-limiting conditions (not just water scarcity).
• Exploring how to incorporate uncertainty quantification into the PI-RNN could provide more robust forecasts and risk assessments.

In summary, the paper presents a promising physics-informed deep learning approach for pixel-level crop yield loss forecasting, demonstrating improved or comparable performance to state-of-the-art ML models while providing valuable, physically consistent intermediate outputs that enhance interpretability for agricultural stakeholders.