We show that suitably regular functions can be approximated in the $\mathcal{C}^1$-norm both with rational functions and rational neural networks, including approximation rates with respect to width and depth of the network, and degree of the rational functions. As consequence of our results, we further obtain $\mathcal{C}^1$-approximation results for rational neural networks with the $\text{EQL}^÷$ and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.

Finally, we use the identity ReLU( x) = |x | + x 2, x R, to define a rational approximation to the ReLU function on the interval [ 1, 1] as r (x) = 1 2 null xr ( x) 1 + null + x null . Therefore, we have the following inequalities for x [ 1, 1], | ReLU( x) r (x) | = 1 2 null null null null | x| xr ( x) 1 + null null null null null 1 2(1 + null) (||x | xr (x) | + null| x |) null 1 + null null. We now show that ReLU neural networks can approximate rational functions. The structure of the proof closely follows [12, Lemma 1.3]. The statement of Theorem 3 comes in two parts, and we prove them separately.

We propose a novel machine learning architecture that departs from conventional neural network paradigms by leveraging quantum spectral methods, specifically Pade approximants and the Lanczos algorithm, for interpretable signal analysis and symbolic reasoning. The core innovation of our approach lies in its ability to transform raw time-domain signals into sparse, physically meaningful spectral representations without the use of backpropagation, high-dimensional embeddings, or data-intensive black-box models. Through rational spectral approximation, the system extracts resonant structures that are then mapped into symbolic predicates via a kernel projection function, enabling logical inference through a rule-based reasoning engine. This architecture bridges mathematical physics, sparse approximation theory, and symbolic artificial intelligence, offering a transparent and physically grounded alternative to deep learning models. We develop the full mathematical formalism underlying each stage of the pipeline, provide a modular algorithmic implementation, and demonstrate the system's effectiveness through comparative evaluations on time-series anomaly detection, symbolic classification, and hybrid reasoning tasks. Our results show that this spectral-symbolic architecture achieves competitive accuracy while maintaining interpretability and data efficiency, suggesting a promising new direction for physically-informed, reasoning-capable machine learning.

Approximation-based spectral graph neural networks, which construct graph filters with function approximation, have shown substantial performance in graph learning tasks. Despite their great success, existing works primarily employ polynomial approximation to construct the filters, whereas another superior option, namely ration approximation, remains underexplored. Although a handful of prior works have attempted to deploy the rational approximation, their implementations often involve intensive computational demands or still resort to polynomial approximations, hindering full potential of the rational graph filters. To address the issues, this paper introduces ERGNN, a novel spectral GNN with explicitly-optimized rational filter. ERGNN adopts a unique two-step framework that sequentially applies the numerator filter and the denominator filter to the input signals, thus streamlining the model paradigm while enabling explicit optimization of both numerator and denominator of the rational filter. Extensive experiments validate the superiority of ERGNN over state-of-the-art methods, establishing it as a practical solution for deploying rational-based GNNs.

The computational cost for inference and prediction of statistical models based on Gaussian processes with Mat\'ern covariance functions scales cubicly with the number of observations, limiting their applicability to large data sets. The cost can be reduced in certain special cases, but there are currently no generally applicable exact methods with linear cost. Several approximate methods have been introduced to reduce the cost, but most of these lack theoretical guarantees for the accuracy. We consider Gaussian processes on bounded intervals with Mat\'ern covariance functions and for the first time develop a generally applicable method with linear cost and with a covariance error that decreases exponentially fast in the order $m$ of the proposed approximation. The method is based on an optimal rational approximation of the spectral density and results in an approximation that can be represented as a sum of $m$ independent Gaussian Markov processes, which facilitates easy usage in general software for statistical inference, enabling its efficient implementation in general statistical inference software packages. Besides the theoretical justifications, we demonstrate the accuracy empirically through carefully designed simulation studies which show that the method outperforms all state-of-the-art alternatives in terms of accuracy for a fixed computational cost in statistical tasks such as Gaussian process regression.

We view the ARMA (Autoregressive Moving Average) model of a stationary process as a random variable approximation. By mapping stationary random variables to Hardy space functions on the unit disk, we can turn a problem of random variable approximation to a newly formulated problem of function approximation. When the functions are continuous, the spectral theorem provides a link between these two points of view, allowing us to provide approximation guarantees for a certain class of stationary processes, and also to identify certain other stationary processes that seem difficult to approximate. We were unable to find similar approximation or approximability guarantees in our review of time series and distributed lags literature. For instance, as detailed in the next section, Box and Jenkins ([BJ76]) assume that a long moving average representation obtained from Wold's decomposition can be mapped to a short autoregressive representation based on some examples and analogies, and provide no specific guarantees for general stationary processes. They tackle the existence of a stable ARMA model using the notion of unit roots.

Rational and neural network based approximations are efficient tools in modern approximation. These approaches are able to produce accurate approximations to nonsmooth and non-Lipschitz functions, including multivariate domain functions. In this paper we compare the efficiency of function approximation using rational approximation, neural network and their combinations. It was found that rational approximation is superior to neural network based approaches with the same number of decision variables. Our numerical experiments demonstrate the efficiency of rational approximation, even when the number of approximation parameters (that is, the dimension of the corresponding optimisation problems) is small. Another important contribution of this paper lies in the improvement of rational approximation algorithms. Namely, the optimisation based algorithms for rational approximation can be adjusted to in such a way that the conditioning number of the constraint matrices are controlled. This simple adjustment enables us to work with high dimension optimisation problems and improve the design of the neural network. The main strength of neural networks is in their ability to handle models with a large number of variables: complex models are decomposed in several simple optimisation problems. Therefore the the large number of decision variables is in the nature of neural networks.

Rational approximations are introduced and studied in granular graded rough sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between granularity, mereology and approximations in the context. In addition, a framework for rational approximations is introduced by her in the mentioned paper(s). Granular approximations constructed as per the procedures of variable precision rough sets (VPRS) are likely to be more rational than those constructed from a classical perspective under certain conditions. This may continue to hold for some generalizations of the former. However, a formal characterization of such conditions is not available in the previously published literature. In this research, theoretical aspects of the problem are critically examined, uniform generalizations of granular VPRS are introduced, new connections with granular graded rough sets are proved, appropriate concepts of substantial parthood are introduced, their extent of compatibility with the framework is accessed, and the framework is extended. Basic assumptions are explained in detail, and additional examples are constructed for readability. Furthermore, meta applications to cluster validation, image segmentation and dynamic sorting are invented. Extensions to direct generalizations of VPRS such as probabilistic rough sets are a natural consequence of the work.