ICML 2025

Closed-form Solutions: A New Perspective on Solving Differential Equations
Shu Wei 1 2 Yanjie Li 1 Lina Yu 1 2 Weijun Li 1 3 4 Min Wu 1 Linjun Sun 1 Jingyi Liu 1 Hong Qin 1 3
Yusong Deng 1 2 Jufeng Han 1 3 Yan Pang 1 4
Abstract
The quest for analytical solutions to differential
equations has traditionally been constrained by
the need for extensive mathematical expertise.
Machine learning methods like genetic algorithms
have shown promise in this domain, but are hin-
dered by significant computational time and the
complexity of their derived solutions. This paper
introduces SSDE (Symbolic Solver for Differen-
tial Equations), a novel reinforcement learning-
based approach that derives symbolic closed-form
solutions for various differential equations. Eval-
uations across a diverse set of ordinary and par-
tial differential equations demonstrate that SSDE
outperforms existing machine learning methods,
delivering superior accuracy and efficiency in ob-
taining analytical solutions.
1. Introduction
Differential equations (DEs) are foundational to mathemat-
ics, physics, and the natural sciences, providing abstract
models for diverse physical phenomena. For example, Pois-
son’s equation governs the distribution of electrostatic poten-
tial, while the heat equation describes temperature variations
within an object (Evans, 2022). Solving these DEs is critical
for understanding and predicting the dynamics of complex
physical systems (Alkhadhr & Almekkawy, 2021; Savovi´c
et al., 2023).
Deriving analytical solutions requires rigorous analysis of
their existence, uniqueness, stability, and behavioral proper-
ties, often leveraging advanced mathematical theories and
1AnnLab, Institute of Semiconductors, Chinese Academy of
Sciences, Beijing, China 2College of Materials Science and Opto-
Electronic Technology, University of Chinese Academy of Sci-
ences, Beijing, China 3School of Integrated Circuits, University
of Chinese Academy of Sciences, Beijing, China 4School of
Industry-ed...