Contemporary economic systems—financial markets, institutional networks, monetary systems—exhibit nonlinear, multiscale, and often unstable behaviors. Classical approaches, based on linear models, simple stochastic differential equations [1], or standard neural networks [2], struggle to capture extreme regimes, phase transitions, long-memory effects [3], and structural breaks.

Kolmogorov-Arnold Networks (KAN), inspired by the Kolmogorov-Arnold theorem [4], offer a powerful alternative: weights are replaced by learned univariate functions, enabling fine, local, and interpretable modeling. When integrated into a reproducing kernel Hilbert space (RKHS), these networks acquire a rigorous geometric structure in which functional complexity is controlled by an associated energy [5].

The use of B-splines with grid extension further strengthens this approach: it allows the capture of extreme behaviors, improves boundary accuracy, and ensures numerical stability essential for economic applications [6].

Financial instability and risk propagation phenomena also motivate such modeling, consistent with the theoretical foundations presented in Bouchaud and Potters [7]. Moreover, earlier work on stochastic differential equations and fractional perturbations provides additional justification for adopting energy-based formulations in monetary instability modeling.

We consider an energy functional:

where the first term measures functional complexity and the second term is a data fitting term. This formulation is consistent with classical energy-based models and stochastic perturbation theory [8]. It also aligns with earlier work on stochastic differential equations and fractional perturbations.

The associated stochastic dynamics is:

Kolmogorov-Arnold Networks can be parameterized using different functional bases. Two families dominate the recent literature: B-splines (used in original KAN) and Chebyshev polynomials (used in ChebyKAN). These two approaches exhibit very different structural properties, influencing numerical stability, interpretability, and approximation capacity.

We consider the refined stochastic formulation of the monetary variable:

where the monetary energy is defined as:

This formulation is consistent with classical stochastic perturbation theory [8] and provides a geometric interpretation of monetary instability. It also aligns with earlier work on stochastic differential equations and fractional perturbations, reinforcing the relevance of energy-based modeling for unstable monetary systems.

The original model, introduced in, is formulated as:

Each drift component corresponds to a KAN activation:

The diffusion term σ 0 corresponds to the stochastic component in the KAN-RKHS-Energy reformulation. This interpretation is consistent with classical stochastic perturbation theory [8] and with the earlier analysis of stochastic differential equations in.

To illustrate how classical monetary models naturally embed into the proposed KAN-RKHS-Energy framework, we revisit the heterogeneous rationality model developed in. The population is composed of rational agents (coherent expectations) and non-rational agents (equiprobable beliefs). The monetary variable M t aggregates these expectations.

Figure 1 summarizes the monetary model in the KAN-RKHS-Energy framework. It highlights the conceptual link between heterogeneous rationality, KAN activations, the monetary energy function, and the emergence of attractors or crises.

Figure 1. Conceptual link between the heterogeneous rationality model and the KAN-RKHS-Energy framework.

Before introducing the figures, we note that the transition from B-splines to 3-splines is illustrated in Figure 2, which highlights the additional flexibility introduced by the tension and bias parameters.

We extend the Hilbert-Energy framework by introducing 3-splines as the functional basis for KAN activations. This preserves the variational structure while improving local flexibility and the ability to model asymmetric ruptures.

Figure 2. Transition from B-spline basis to 3-spline basis.

Figure 3.Effect of tension parameter β 1 : higher β 1 yields sharper curvature.

Figure 4. Effect of bias parameter β 2 : local asymmetry of the response.

Figure 5. Monetary model as a special case of the KAN-B-splines/3-splines RKHS framework.

Figure 2 illustrates the transition from B-splines to 3-splines. Figure 3 shows the effect of the tension parameter β 1 . Figure 4 shows the effect of the bias parameter β 2 . Figure 5 presents the monetary model as a special case of the KAN-RKHS-Energy framework.

We simulate the monetary dynamics over 100 monthly steps using the parameters:

The statistical evaluation of the fitted monetary model yields:

All drift coefficients satisfy p < 0.05 , indicating statistical significance of the estimated functional components. The high R 2 confirms that the KAN-RKHS-Energy formulation captures the main features of the observed monetary dynamics.

These results validate the relevance of combining KAN activations, RKHS geometry, and energy-based modeling for unstable monetary systems.

We have presented a general framework combining Kolmogorov-Arnold Networks, reproducing kernel Hilbert spaces, and an energy-based formulation for modeling complex systems. This framework makes it possible to represent nonlinear dynamics in an interpretable, stable manner compatible with the analytical requirements of economic systems.

The integration of B-splines with grid extension plays a central role: it captures extreme behaviors, improves boundary accuracy, and provides a multiscale structure adapted to real economic phenomena.

The extension using 3-splines further strengthens this approach by introducing controlled asymmetry and local refinement of the energy structure, particularly suited to unstable monetary systems and regime transitions. The application to the Malagasy monetary instability model illustrates the power of this formulation: the model, reformulated in an RKHS, reveals an underlying energy structure that identifies attractors, tension zones, and rupture points.

This functional reading opens the way to finer analyses of emerging monetary systems, especially in contexts of institutional vulnerability or external pressures. This work thus establishes a solid foundation for modern economic modeling, unifying functional theory, interpretable machine learning, and energy analysis.

import numpy as np

import matplotlib. pyplot as plt

T = 20

gamma = 0.184

c_H, c_L = 10, 5

p_r = 0.7

p_i = 0.5

states_r = np.random.binomial (1, p_r, T)

states_i = np. random. binomial (1, p_i, T)

c_r = np. where (states_r == 1, c_H, c_L)

c_i = np. where (states_i == 1, c_H, c_L)

M = gamma * c_r + (1 - gamma) * c_i

plt. plot (M, label = “Aggregate monetary variable”)

plt. plot (c_r, '--', label = “Rational agents”)

plt. plot (c_i, '--', label = “Non-rational agents”)

plt. legend ( )

plt. grid (True)

plt. show ( )