TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansion

Partial differential equations (PDEs) underpin many physical phenomena, but traditional numerical solvers face computational bottlenecks, especially in high-dimensional, multi-scale systems. Recent advances in operator learning—such as DeepONet and Fourier Neural Operator—have shown promise by leveraging neural networks to approximate solution operators directly from data. These methods, however, typically rely on fixed temporal discretization, which limits their ability to adapt to the uneven temporal complexity inherent in real-world systems. During rapid changes, fixed steps lead to error accumulation and inefficiency, while during slow evolution, they waste computational resources.

To address these challenges, this paper introduces TANTE (Time-Adaptive Transformer with Neural Taylor Expansion), a novel framework that enables continuous-time prediction with dynamically adjustable step sizes. The core idea is to use neural networks to learn high-order temporal derivatives and the local radius of convergence, allowing the model to perform a Taylor expansion at each step. This approach provides a flexible, data-driven mechanism to adapt the prediction interval based on the local behavior of the solution, reducing cumulative errors and improving efficiency.

TANTE's architecture comprises three main components: a spatiotemporal encoder, which extracts features from input data and modulates them with temporal information; a Transformer processor, which estimates derivatives along different axes; and a decoder, which predicts derivatives and estimates the convergence radius. The model then sums the Taylor series within the estimated interval for continuous-time forecasts. When predictions extend beyond this interval, the model switches to autoregressive mode, incorporating previous predictions.

Extensive experiments across four challenging PDE benchmarks—Rayleigh-Bénard convection, active matter, viscoelastic fluids, and turbulent radiative layers—demonstrate TANTE's superior performance. The results show a 60-80% reduction in prediction error and a 30-40% speed-up in inference compared to fixed-step baselines. The model also exhibits robust stability and adaptability, effectively handling uneven temporal dynamics. These advances open new possibilities for efficient, accurate modeling of complex physical systems, with potential impacts spanning weather prediction, fluid mechanics, materials science, and beyond. Despite its strengths, challenges remain in handling noisy data, extreme nonlinearities, and high-dimensional problems, guiding future research directions toward further robustness and scalability.