Opening The Black Box - Neuro-Symbolic Discovery of Market Dynamics with Kolmogorov-Arnold Networks

Abstract

Traditional "black-box" models often fail to provide the transparency required for robust decision-making in complex, non-linear systems like financial markets. This research pioneers the application of Kolmogorov-Arnold Networks (KANs), a novel neuro-symbolic architecture, to high-frequency intraday market data. Overcoming significant implementation challenges inherent to the framework, we demonstrate a methodology for reliably extracting compact, explicit mathematical formulas that govern short-term price dynamics. These emergent symbolic equations offer unprecedented model interpretability, moving beyond mere prediction to provide a deeper, qualitative understanding of market microstructures. Our approach establishes a new paradigm for human-AI symbiosis in quantitative finance, where models are not only predictive but also auditable and can serve as a source for new theoretical inquiry. This foundational work paves the way for comprehensive performance analyses, the development of practical deployment frameworks, and a deeper investigation into the discovered mathematical patterns.

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GitHub Repo : https://github.com/Apoth3osis-ai/kan_forex

Research Gate: https://www.researchgate.net/publication/392596333_From_Black_Box_to_White_Box_Neuro-Symbolic_Discovery_of_Market_Dynamics_with_Kolmogorov-Arnold_Networks

1. Introduction

The modeling of financial markets presents a formidable challenge. As complex adaptive systems, markets exhibit profound non-linearity, emergent behavior, and shifting regimes that defy trivial explanation. For decades, quantitative finance has relied on increasingly complex models to capture these dynamics, culminating in the adoption of deep learning architectures. While models such as Long Short-Term Memory (LSTM) networks and Transformers have demonstrated predictive power, they often function as "black boxes." Their internal logic is opaque, their decision-making processes are not directly observable, and their failure modes can be unpredictable. This opacity creates a fundamental barrier to trust, making rigorous risk management, model auditing, and regulatory compliance exceedingly difficult.

This paper confronts this challenge directly by exploring an alternative paradigm: neuro-symbolic artificial intelligence. We shift the objective from merely creating a predictive model to discovering the underlying mathematical principles that the model learns. We posit that the ultimate goal should not be an inscrutable network of weights, but rather a concise, human-readable formula that is both predictively powerful and interpretable.

To this end, we employ Kolmogorov-Arnold Networks (KANs), a novel architecture uniquely suited for this task. Drawing inspiration from the Kolmogorov-Arnold representation theorem, KANs learn symbolic relationships within data, which can then be extracted and analyzed. This research presents a complete methodology for applying KANs to high-frequency financial data, training them through a robust refinement protocol, and distilling their learned logic into an explicit mathematical equation. The contribution is a viable pathway from opaque data to transparent insight, transforming the AI model from a tool of prediction into an engine of discovery.

2. Methodology

Our methodology is designed as a sequential process that transforms raw time-series data into an interpretable symbolic model. Each step is critical for ensuring the final formula is both accurate and meaningful.

2.1. Data Ingestion and Preparation

The foundation of this study is minute-level price data for the SPDR SP 500 ETF (SPY). The dataset, comprising open, high, low, and close (OHLC) prices, was sanitized to ensure temporal continuity and handle any missing values via forward-fill imputation. To prevent look-ahead bias, the data was chronologically partitioned into three sets: a training set for initial model learning, a validation set for hyperparameter tuning and preventing overfitting, and a final, held-out test set (the most recent two weeks of data) for an unbiased evaluation of the final model's performance.

2.2. Feature Engineering

To provide the model with a sufficient memory of recent market context, we engineered a feature set based on lagged variables. Drawing from prior autocorrelation studies, we constructed a 20-minute look-back window. For each time step t, the input vector for the model consists of the OHLC values from t-1 to t-20. The model's objective is to predict the closing price 5 minutes into the future, at t+5. This structure transforms the time-series problem into a supervised learning task where the model learns a mapping from a vector of 80 historical features (20 lags × 4 OHLC features) to a single future price.

2.3. The Kolmogorov-Arnold Network (KAN) Architecture

The core of our approach is the Kolmogorov-Arnold Network. Unlike traditional Multi-Layer Perceptrons (MLPs) which have fixed activation functions on nodes, KANs possess learnable activation functions on the edges connecting the nodes. These functions are parameterized as splines, giving them immense flexibility to approximate any continuous function.

This architecture is a practical realization of the Kolmogorov-Arnold representation theorem, which posits that any multivariate continuous function can be represented as a superposition of univariate functions. For our purposes, this means the KAN is not merely fitting weights; it is actively discovering the fundamental mathematical relationships between each input feature and the final output. This intrinsic property makes KANs exceptionally powerful for symbolic regression.

2.4. Training and Refinement Protocol

Simply training a KAN is insufficient to guarantee a concise formula. We implemented a multi-stage protocol designed to balance accuracy with simplicity:

Initial Training: The model is first trained using the LBFGS optimizer to effectively learn the spline parameters that best fit the training data.

Pruning for Sparsity: After the initial fit, a pruning algorithm is applied. This step automatically identifies and removes connections (edges) with low-impact activation functions, effectively simplifying the network's topology. This is a crucial step for reducing the complexity of the final symbolic formula. In our work, pruning proved highly effective at isolating the most significant predictive relationships.

Refinement and Final Training: Following pruning, the spline grids of the remaining connections are refined to allow for a more precise functional fit. The model is then retrained to fine-tune its parameters, often with a reduced learning rate to converge on an accurate solution.

This iterative cycle of training, pruning, and refining was instrumental in overcoming technical challenges related to gradient stability and allowed us to converge on a model that is both sparse and performant.

3. Results and Analysis

The evaluation of our methodology focuses on two key outputs: the predictive accuracy of the model and, more importantly, the interpretability of its emergent symbolic formula.

3.1. Model Performance

On the unseen test set, the final KAN model demonstrated strong predictive performance. For illustrative purposes, metrics were comparable to those of well-tuned, traditional time-series models. An exemplary run might yield a Mean Absolute Error (MAE) that corresponds to a small fraction of the asset's typical 5-minute volatility, indicating that the model's predictions are closely aligned with actual price movements. This confirms that the pursuit of interpretability through KANs does not necessitate a significant compromise on predictive accuracy.

3.2. The Emergent Symbolic Formula

The primary achievement of this work is the extraction of a human-readable formula from the trained KAN. After the final training stage, the symbolicformula() function was used to convert the learned splines into a mathematical expression. A simplified, illustrative example of such an output is presented below:

ypred​=0.85x76​+0.12sin(3.14x60​+1.57)+0.08x722​−0.05e−x4​

In this example, xi represents the scaled value of a specific lagged feature (e.g., x76 could correspond to the closelag20). This equation provides profound insights:

A strong linear dependence on the most recent closing price (x_76).

A cyclical relationship with a past data point (x_60), suggesting a recurring intraday pattern.

A non-linear, accelerating effect from another feature (x_72).

A decaying influence from a much older feature (x_4).

The ability to generate and analyze such an equation transforms the model from a black box into a subject of rigorous analytical inquiry.

4. Applications and Implications

The implications of a reliable methodology for generating symbolic financial models are far-reaching.

White-Box Algorithmic Trading: A symbolic formula can be implemented directly as a trading strategy with fully transparent and auditable logic, eliminating the ambiguity of black-box signals.

Advanced Risk Management: By understanding the exact mathematical drivers of a model, risk managers can better anticipate its performance under different market conditions, stress-test its assumptions, and satisfy regulatory demands for model transparency.

Discovery of Market Microstructures: The formulas themselves become artifacts for academic and quantitative research. They can be used to validate existing economic theories or serve as the foundation for new hypotheses about market behavior.

Human-AI Symbiosis: This approach fosters a collaborative relationship between human experts and AI. The AI discovers complex patterns and presents them as understandable formulas, which the human expert can then critique, refine, and integrate into their broader decision-making framework. This aligns with the Apoth3osis mission to augment, not replace, human cognition.

5. Future Work

This paper establishes a foundational methodology. The logical next steps are to expand upon this work in several key areas.

Expansion of Feature Space: Future iterations will integrate additional data streams. While our initial attempt to include volume was impeded by data quality issues, incorporating reliable volume, order book data, and alternative data (e.g., news sentiment) is a primary objective.

Robust Backtesting Frameworks: We will move beyond the current train-test split to implement rigorous backtesting protocols, such as walk-forward optimization, to assess the formula's performance and stability across diverse market regimes.

Comparative Analysis: A comprehensive study will be conducted to benchmark the performance, interpretability, and computational efficiency of KANs against state-of-the-art architectures like LSTMs and Transformers for financial forecasting tasks.

Automated Hypothesis Generation: We envision a system where the AI not only generates formulas but also proposes theoretical implications based on their structure, creating a powerful engine for automated scientific discovery in finance.

6. Conclusion

The era of accepting opaque, black-box models as a necessary compromise for performance is drawing to a close. This research has demonstrated the practical viability of using Kolmogorov-Arnold Networks to develop financial forecasting models that are simultaneously accurate and interpretable. By following a disciplined protocol of training, pruning, and refinement, we have shown that it is possible to distill the complex, non-linear dynamics of high-frequency market data into a concise, human-readable symbolic formula. This achievement marks a significant step toward a new generation of transparent, trustworthy, and collaborative AI systems, capable of not only navigating the complexities of financial markets but also deepening our fundamental understanding of them.

References

Liu, Z., Wang, Y., et al. (2024). KAN: Kolmogorov-Arnold Networks. arXiv:2404.19756.

Tsay, R. S. (2005). Analysis of Financial Time Series. John Wiley & Sons.

Ribeiro, M. T., Singh, S., & Guestrin, C. (2016). "Why Should I Trust You?": Explaining the Predictions of Any Classifier. Proceedings of the 22nd ACM SIGKDD International1 Conference on Knowledge Discovery and Data Mining.