Theory and Learning
for Complex Systems

Yield curves are dynamic functions indexed by maturity and observed through irregular market conventions. Accurate forecasting therefore requires both temporal modeling and a representation that respects the functional structure of the curve. This paper presents Gaussian Process Operator Regression (GPOR), a probabilistic operator-learning framework for multi-horizon yield curve forecasting. GPOR treats each historical yield curve as a function on log-maturity, constructs sensor representations from lagged curves, and learns a map from these representations to future curve increments. A sparse variational Gaussian process with a separable kernel provides scalable inference and predictive intervals, while a mean-reversion regularizer encourages economically plausible behavior at longer horizons. Experiments on daily U.S. Treasury par yield curves evaluate forecast horizons of 1, 5, 10, 15, and 20 trading days against dynamic Nelson–Siegel variants, factor models, and deep learning models. GPOR is competitive across horizons, with stronger RMSE performance from five trading days onward and the lowest relative L2 errors across all evaluated horizons. It also remains usable when evaluated on denser, sparser, and off-grid maturity sets. All stochastic experiments are repeated over multiple random seeds and reported as mean ± standard deviation. Ablation and sensitivity studies show that lagged functional sensors, nonlinear sensor placement, and mean-reverting regularization are important to the model’s robustness. The results suggest that Gaussian-process operator learning is a promising uncertainty-aware approach for financial spatiotemporal forecasting.

Two aspects of the probabilistic approach to classification by deep neural networks are investigated: the distribution of approximation errors of functions drawn from a given probability and their learnability from random samples of data. We show that the crucial factor influencing the behavior of error functionals is the growth of the sets of network input-output functions as the size of the data set increases. Its polynomial growth implies that both approximation and empirical errors concentrate around their mean values. The concentration of empirical errors determined by data samples is beneficial for their convergence to the theoretical error. However, the tight concentration of approximation errors need not be beneficial because in the case of large mean error it means that almost all functions cannot be accurately computed. Influence of the polynomial growth of sets of input-output functions is analyzed for the class of deep ReLU networks.

While Vision Transformers (ViTs) often exhibit better calibration than CNNs, we observed they still suffer from overconfidence in supervised training. We hypothesize this is tied to the unbounded nature of Cross-Entropy coupled with Softmax. We explore a geometric equivalence between temperature-scaled Softmax and the Gaussian density function, introducing a scale-agnostic framework: Pure Spatial Regression. By projecting features onto a normalized hypersphere and modulating the objective via the ratio of LayerNorm gain to internal spatial dispersion (γ/B), the model structurally balances feature sharpness and calibration, significantly improving accuracy and Expected Calibration Error (ECE).

We introduce a chaos-informed spatio-temporal dataset fingerprinting framework for characterizing the predictability of real-world forecasting datasets before model selection. The framework represents each dataset using compact descriptors of temporal structure, spatial coherence, empirical predictability, and signal complexity, and calibrates these descriptors in a controlled chaotic-system space. We use coupled standard map (CSM) synthetic regimes from ChaosNetBench (CNB) as the calibration set, where local nonlinearity, coupling strength, system size, and topology are known. We demonstrate the framework on the METR-LA and PEMS-BAY benchmark traffic datasets by computing raw and seasonally residualized fingerprints and projecting them into the CSM space. Both datasets remain separated from the CNB calibration set, but the signed descriptor gaps reveal that calendar structure contributes differently to the two fingerprints: residualization reduces the separation distance for both datasets, much more strongly for PEMS-BAY than for METR-LA. This descriptor contrast aligns qualitatively with a compact forecasting response comparison using TCN and LSTM as temporal baselines and Graph WaveNet as the STGNN comparator. The dataset with the closer residual spatial fingerprint also shows stronger relative response to Graph WaveNet. The proposed framework provides a chaos-informed way to compare observable spatio-temporal dataset fingerprints and support prediction-model selection and analysis.