In the area of explainable artificial intelligence, Symbolic Regression (SR) has emerged as a promising approach by discovering interpretable mathematical expressions that fit data. However, SR faces two main challenges: most methods are evaluated on scientific datasets with well-understood relationships, limiting generalization, and SR primarily targets single-output regression, whereas many real-world problems involve multi-target outputs with interdependent variables. To address these issues, we propose multi-task regression GINN-LP (MTRGINN-LP), an interpretable neural network for multi-target symbolic regression. By integrating GINN-LP with a multi-task deep learning, the model combines a shared backbone including multiple power-term approximator blocks with task-specific output layers, capturing inter-target dependencies while preserving interpretability. We validate multi-task GINN-LP on practical multi-target applications, including energy efficiency prediction and sustainable agriculture. Experimental results demonstrate competitive predictive performance alongside high interpretability, effectively extending symbolic regression to broader real-world multi-output tasks.

This paper presents the use of Kolmogorov-Arnold Networks (KANs) for forecasting the CBOE Volatility Index (VIX). Unlike traditional MLP-based neural networks that are often criticized for their black-box nature, KAN offers an interpretable approach via learnable spline-based activation functions and symbolification. Based on a parsimonious architecture with symbolic functions, KAN expresses a forecast of the VIX as a closed-form in terms of explanatory variables, and provide interpretable insights into key characteristics of the VIX, including mean reversion and the leverage effect. Through in-depth empirical analysis across multiple datasets and periods, we show that KANs achieve competitive forecasting performance while requiring significantly fewer parameters compared to MLP-based neural network models. Our findings demonstrate the capacity and potential of KAN as an interpretable financial time-series forecasting method.

Hydrological models often involve constitutive laws that may not be optimal in every application. We propose to replace such laws with the Kolmogorov-Arnold networks (KANs), a class of neural networks designed to identify symbolic expressions. We demonstrate KAN's potential on the problem of baseflow identification, a notoriously challenging task plagued by significant uncertainty. KAN-derived functional dependencies of the baseflow components on the aridity index outperform their original counterparts. On a test set, they increase the Nash-Sutcliffe Efficiency (NSE) by 67%, decrease the root mean squared error by 30%, and increase the Kling-Gupta efficiency by 24%. This superior performance is achieved while reducing the number of fitting parameters from three to two. Next, we use data from 378 catchments across the continental United States to refine the water-balance equation at the mean-annual scale. The KAN-derived equations based on the refined water balance outperform both the current aridity index model, with up to a 105% increase in NSE, and the KAN-derived equations based on the original water balance. While the performance of our model and tree-based machine learning methods is similar, KANs offer the advantage of simplicity and transparency and require no specific software or computational tools. This case study focuses on the aridity index formulation, but the approach is flexible and transferable to other hydrological processes.

Neuro-Symbolic (NeSy) integration combines symbolic reasoning with Neural Networks (NNs) for tasks requiring perception and reasoning. Most NeSy systems rely on continuous relaxation of logical knowledge, and no discrete decisions are made within the model pipeline. Furthermore, these methods assume that the symbolic rules are given. In this paper, we propose Deep Symbolic Learning (DSL), a NeSy system that learns NeSy-functions, i.e., the composition of a (set of) perception functions which map continuous data to discrete symbols, and a symbolic function over the set of symbols. DSL learns simultaneously the perception and symbolic functions while being trained only on their composition (NeSy-function). The key novelty of DSL is that it can create internal (interpretable) symbolic representations and map them to perception inputs within a differentiable NN learning pipeline. The created symbols are automatically selected to generate symbolic functions that best explain the data. We provide experimental analysis to substantiate the efficacy of DSL in simultaneously learning perception and symbolic functions.

Systematic compositionality is an essential mechanism in human language, allowing the recombination of known parts to create novel expressions. However, existing neural models have been shown to lack this basic ability in learning symbolic structures. Motivated by the failure of a Transformer model on the SCAN compositionality challenge (Lake and Baroni, 2018), which requires parsing a command into actions, we propose two auxiliary sequence prediction tasks that track the progress of function and argument semantics, as additional training supervision. These automatically-generated sequences are more representative of the underlying compositional symbolic structures of the input data. During inference, the model jointly predicts the next action and the next tokens in the auxiliary sequences at each step. Experiments on the SCAN dataset show that our method encourages the Transformer to understand compositional structures of the command, improving its accuracy on multiple challenging splits from <= 10% to 100%. With only 418 (5%) training instances, our approach still achieves 97.8% accuracy on the MCD1 split. Therefore, we argue that compositionality can be induced in Transformers given minimal but proper guidance. We also show that a better result is achieved using less contextualized vectors as the attention's query, providing insights into architecture choices in achieving systematic compositionality. Finally, we show positive generalization results on the groundedSCAN task (Ruis et al., 2020). Our code is publicly available at: https://github.com/jiangycTarheel/compositional-auxseq