Learning Parametric Nitrogen Fertilizer Response Curves Using Neuro Symbolic Regression

Technical Summary: Learning Parametric Nitrogen Fertilizer Response Curves Using Neuro Symbolic Regression

Problem Statement
Accurately modeling crop response to Nitrogen (N) fertilization is a critical challenge in precision agriculture (PA), directly impacting economic returns and environmental sustainability. Current methodologies face a dichotomy: they either rely on predefined parametric forms (e.g., quadratic-plateau or exponential functions) that may not capture complex, site-specific dynamics, or they utilize opaque machine learning models (such as deep neural networks) that lack interpretability. Furthermore, agricultural data is often sparse and subject to epistemic uncertainty, making it difficult to discover robust functional relationships without overfitting to noise or localized data gaps. The specific goal of this work is to distill parametric, site-specific N-response curves from opaque models without assuming a predefined functional form, thereby enabling the discovery of mathematical equations that characterize the relationship between N-fertilizer rates and crop yield within specific Management Zones (MZs).

Methodology
The authors propose a Neuro Symbolic Regression (SR) framework based on SeTGAP (Symbolic Regression using Transformers, Genetic Algorithms, and Genetic Programming). The approach is decomposed into two primary phases:

1. Multi-Set Symbolic Skeleton Prediction (MSSP):
   To address epistemic uncertainty and data sparsity, the method employs a Multi-Set Transformer strategy. Instead of training on a single dataset, the algorithm constructs a collection of distinct input subsets ($D = \{D^{(1)}, \dots, D^{(N_S)}\}$) by randomly sampling subregions from the input domain.

   • 1-D Synthetic Validation: In the initial phase, the method generates multiple subsets from a single 1-D curve. The Multi-Set Transformer, pre-trained on a corpus of symbolic expressions, predicts a "symbolic skeleton" (an abstract representation with constant placeholders) that captures the shared functional structure across these subsets. This diversity helps the model ignore localized noise and recover the underlying functional form.
   • Real-World Application: For the winter wheat case study, the strategy is adapted to handle multiple N-response curves within a specific Management Zone ($R_z$). The algorithm selects $N_S$ distinct curves from the zone, samples subsets from each, and generates candidate skeletons that generalize across the varying curves within that zone.

2. Parameter Fitting and Optimization:
   Once a symbolic skeleton is identified, the placeholder constants are fitted to the observed data using a Genetic Algorithm (GA). The process iterates to minimize the Mean Squared Error (MSE) between the fitted expression and the target values. The final output is a compact, interpretable mathematical equation.

Key Contributions

• Neuro-Symbolic Framework for PA: The paper introduces a novel application of SeTGAP to extract parametric N-response curves from opaque Neural Network (NN) predictions, moving beyond black-box predictions to interpretable mathematical models.
• Multi-Set Strategy for Uncertainty: A specific contribution is the adaptation of MSSP to handle epistemic uncertainty. By enforcing consistency across diverse input subsets (or distinct curves within an MZ), the method recovers robust skeletons even in data-scarce regimes, preventing the overfitting of noise-driven anomalies.
• Zone-Specific Functional Discovery: The work demonstrates the feasibility of learning distinct functional forms for different Management Zones within a single field, challenging the traditional assumption that a single parametric form applies to an entire field.

Experimental Results
The study evaluates the approach in two stages:

1. Synthetic 1-D Problems: The method was tested on synthetic problems (cos, hetero, cosqr) with heteroscedastic noise under an Adaptive Sampling (AS) framework. Results showed that while early iterations with sparse data yielded unstable or overfitted expressions, the symbolic models stabilized and converged to the correct functional forms (e.g., identifying $10 - 5\sin(x)$) as the AS process reduced uncertainty. The method successfully recovered correct expressions even when the underlying NN predictions exhibited local inaccuracies due to sparse data.
2. Winter Wheat Case Study (Field A): The method was applied to real-world data from a winter wheat field divided into four MZs. The discovered symbolic expressions were compared against traditional quadratic-plateau and exponential models.
   • Accuracy: The SeTGAP-derived expressions achieved lower Mean Squared Errors (MSE) than traditional models across all four MZs. For example, in MZ 2, the SeTGAP error was 0.0355 compared to 0.0725 for the quadratic-plateau model.
   • Diversity: The method successfully identified different functional forms for different zones (e.g., a hyperbolic tangent form for MZ 1 and 2, a cosine-modulated linear form for MZ 3, and an exponential-sine form for MZ 4), reflecting the spatial variability of soil and terrain.

Significance and Claims
The authors claim that this work demonstrates the potential of neuro-symbolic regression to enable the discovery of site-specific agronomic relationships. By replacing computationally expensive NNs with lightweight, interpretable equations, the framework supports efficient downstream multi-objective optimizations, such as calculating the Economic Optimum Nitrogen Rate (EONR).

The paper modestly frames its contribution as a proof of concept. It acknowledges that the proposed SR strategy is an "ad hoc solution" tailored to the specific challenges of N-response curve distillation rather than a general state-of-the-art SR methodology. The authors emphasize that while the results on Field A show practical potential, future work is required to generalize the strategy to higher-dimensional settings, formalize the methodology, and validate its robustness across multiple fields, farms, and crop types. The primary value lies in providing a data-driven, interpretable alternative to opaque models that captures complex local variations without pre-defining functional forms.