AgriPINN: Process-Informed Biomass Predictor

• AgriPINN is a process-informed neural network that embeds the LINTUL5 biomass accumulation equation into its loss function to ensure physiologically consistent predictions.
• It uses a compact 2D CNN architecture with differentiable physics loss, resulting in significant RMSE reductions and an 8× speedup over conventional models.
• The model jointly retrieves latent variables such as LAI, RUE, and water-stress factor, offering interpretable outputs for precision irrigation, yield forecasting, and climate adaptation.

AgriPINN is a process-informed neural network designed for interpretable and scalable prediction of above-ground biomass (AGB) in major crops under conditions of water stress. It integrates biophysical principles from conventional process-based crop models into a deep learning architecture, employing a differentiable constraint based on a crop-growth ordinary differential equation (ODE). This hybridization enforces physiological consistency in model outputs while achieving the scalability and computational efficiency of modern convolutional neural networks (CNNs). AgriPINN is developed and assessed for winter wheat and maize across Germany, leveraging both extensive historical simulation data and field experiments with controlled water treatments (Shi et al., 22 Jan 2026).

1. Biophysical Model Integration and Loss Formulation

AgriPINN embeds the AGB accumulation equation from the LINTUL5 model as a differentiable residual within its loss function. The discrete evolution for site $p$ and day $t$ is given by: $$AGB(p,t+1) = AGB(p,t) + RUE(p,t) \times I_{\rm int}(p,t) \times F_W(p,t),$$ where $RUE$ is radiation use efficiency, $I_{\rm int}$ is intercepted photosynthetically active radiation (PAR), and $F_W$ is a normalized water-stress factor. Expressing intercepted PAR in terms of LAI, the process equation becomes: $$\Delta AGB(p,t) = RUE(p,t) \cdot PAR(p,t) \cdot [1 - e^{-k\,LAI(p,t)}] \cdot F_W(p,t).$$ AgriPINN enforces this process law via a soft residual penalty in the training objective: $$r(p,t) = \Delta\widehat{AGB}(p,t) - RUE(p,t) \cdot PAR(p,t) \cdot [1 - e^{-k\,LAI(p,t)}] \cdot F_W(p,t).$$ The joint loss is

$$L(\theta) = L_{\rm data}(\theta) + \lambda L_{\rm phys}(\theta),$$

with data loss $L_{\rm data}$ as mean squared error on observed AGB, and physics loss $L_{\rm phys}$ as mean squared residual over a minibatch. Both observational and latent physiologically meaningful quantities are predicted by the network; the ODE constraint is back-propagated as a differentiable regularizer.

2. Convolutional Neural Network Backbone and Data Inputs

The network backbone consists of a compact 2D CNN architecture:

• Inputs: At each spatio-temporal coordinate $(p,t)$, the model ingests daily climate variables (solar radiation, Tmin/Tmax, precipitation), static soil properties, and management actions (fertilizer, irrigation).
• Structure: Three 2D convolutional layers (kernel 5 × 5) with ReLU nonlinearity are followed by pooling, feature flattening, and a fully connected layer producing both observed (AGB) and latent state outputs ($\{LAI, RUE, PAR, F_W\}$).
• Implementation: The model is trained in PyTorch using Adam optimizer with initial LR = 0.001 and cosine-annealing, batch size 128, and $0.3$ dropout; the physics loss weight $\lambda$ is tuned on holdout validation.
• Scalability: Pointwise inference supports arbitrary spatio-temporal densities. Downscaling (ERA5 to 250 m) leverages MODIS NDVI; inference requires a single forward pass, yielding an 8-fold computational speedup over LINTUL5.

3. Recovery of Physiological Latent States

AgriPINN jointly predicts four latent variables associated with crop physiology:

1. Leaf Area Index (LAI)
2. Absorbed photosynthetically active radiation ($I_{\rm int}$)
3. Radiation Use Efficiency (RUE)
4. Water-stress factor ($F_W$)

These latent variables are not directly supervised; only AGB observations inform the data loss. The process-informed physics loss compels the network to select internal states consistent with biophysical dynamics. This design yields physiologically meaningful, unsupervised retrieval of latent processes, supporting both model interpretability and potential use for counterfactual stress analyses.

4. Training Regime and Data Sources

The model is pretrained on 60 years (1951–2015) of synthetic AGB time series simulated by the SIMPLACE (LINTUL5) platform for winter wheat and maize across 397 NUTS-3 regions in Germany. Input data comprises daily meteorology, soil types and textures, and management inputs. The pretraining dataset is split 80%/20% for training and validation, totaling 24,375 region–year samples. Fine-tuning is performed using in-situ measurements from TERENO Selhausen (2016–2018), covering winter wheat and maize under three water treatments (shelter, rainfed, irrigated). Training resumes with reduced learning rate for up to 1000 iterations, maintaining the composite loss formulation.

5. Empirical Performance and Computational Efficiency

Table 1. Pretraining Performance (Historical AGB)

Model	Wheat RMSE	Maize RMSE	Wheat CC	Maize CC
CNN-Transformer	2.08	2.23	0.78	0.72
SLTF	1.43	2.06	0.88	0.81
ConvLSTM-ViT	1.62	1.93	0.82	0.87
AgriPINN	1.42	1.81	0.94	0.92

Table 2. Fine-Tuning Performance (Winter Wheat RMSE, 2016–18)

Treatment	LINTUL5	AgriPINN	ConvLSTM-ViT	SLTF	CNN-Trans.
Shelter	1.92	1.83	2.06	4.47	4.61
Rainfed	4.40	2.25	6.82	9.39	9.45
Irrigated	3.06	2.03	5.88	8.85	8.53

AgriPINN achieves RMSE reductions up to 49% versus LINTUL5 and over 43% versus deep-learning baselines. Paired Wilcoxon tests validate statistical significance ($p<0.01$). Computationally, AgriPINN requires only ≈90 min training time (0.9 M parameters), compared to up to ≈250 min and 12.3 M parameters for large data-driven baselines. Inference speed is 8 × faster than LINTUL5 simulations.

6. Interpretability, Generalization, and Applications

AgriPINN exhibits semantic alignment: elasticities of predicted latent states reflect those in the LINTUL5 process equation, yielding mechanistically grounded explanations. The enforced biophysical constraint also regularizes the hypothesis class, mitigating distribution shift effects. Empirical results show AgriPINN's lower performance degradation between pretraining and field domains compared to black-box neural models.

Key applications include:

• Irrigation Planning: Inferred water-stress factors ($F_W$) and soil moisture signals inform precision irrigation scheduling.
• Yield Forecasting: Predicted daily AGB trajectories support end-of-season yield estimates with greater resilience to weather extremes.
• Climate Adaptation: The model’s interpretable latent variables facilitate scenario analysis under novel meteorological regimes.

7. Limitations and Prospective Extensions

Principal limitations include dependence on the fidelity of the underlying biophysical law (LINTUL5); systematic ODE errors will propagate into the neural network outputs. AgriPINN’s current formulation is tailored to scenarios of water stress, omitting other key stressors such as nutrient limitation or pest pressure.

Anticipated future work involves:

1. Integration of nutrient-cycling and multi-stress modeling (N–P–K constraints).
2. Multi-objective training that merges biomass, yield, and soil moisture outcomes.
3. Incorporation of Bayesian weights or dropout-based confidence intervals for uncertainty quantification.
4. Adaptation for edge deployment to support real-time, high-resolution digital-twin monitoring.

AgriPINN illustrates a robust fusion of process-based crop modeling and deep learning, providing scalable, interpretable, and computationally efficient spatio-temporal biomass predictions under water stress, with prospects for further generalization to broader agricultural decision-support contexts (Shi et al., 22 Jan 2026).