Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis
2026-09-07 12:00Science🔥 40.2 heat score
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A research paper published on September 7, 2026, proposes a framework for data-driven learning using functional analysis theory to solve the problem of solving unknown nonlinear differential equations. This study applies mathematical tools from functional analysis to the field of machine learning, using specific function spaces to represent the dynamic characteristics of unknown systems. The method does not rely on pre-known model parameters or equation forms; instead, it directly infers the structure of the differential equations that control the evolution of unknown nonlinear systems from observed data. The paper demonstrates the effectiveness of this framework in dealing with high-dimensional, complex, and mechanically unknown dynamic systems, providing new theoretical perspectives and algorithmic foundations for scientific computing and physical information fusion.
This paper proposes a interpretable machine learning method based on functional analysis and operator theory, aimed at learning the unknown vector field of nonlinear dynamic systems from single state trajectory data. This method constructs the cost function as the integral distance between two functions in the function space, differing from existing methods that use discrete error sums. An incremental learning algorithm is also proposed for online processing of new samples. This method can detect unknown external forces and underlying dynamics in forced and non-self-consistent (time-varying) dynamic systems, and its advantages are verified through numerical experiments.