用神经算子突破传统微分方程发现局限，能自动找出含非局部项的复杂方程。
A Neural Operator-Based Approach to Symbolic Discovery of PDEs
- 将预训练神经算子作为符号网络节点，构建可微分的方程候选图
- 在含非局部算子、辅助场耦合和记忆积分项的问题上成功恢复简洁方程
- 适合需要发现复杂物理规律的研究者，尤其擅长处理非局部动力学
从数据中发现控制方程仍具挑战性，尤其当底层动力学涉及非局部微分算子、由辅助方程支配的场相互作用或时间记忆效应时。我们提出神经算子驱动的符号模型近似与发现框架（NOMTO），通过在符号网络中引入预训练神经算子作为节点，扩展了类方程学习器的架构。NOMTO将候选方程表示为结合代数运算与固定神经算子代理的稀疏可微计算图，这些神经算子预先训练以逼近非线性算子。我们在包含非局部空间算子、由辅助场方程介导的耦合以及表示记忆效应的时间积分项的模型发现任务上评估该方法。结果表明，NOMTO能够恢复包含非局部算子项的紧凑控制方程，从而将符号模型发现拓展至仅限于局部导数和点对点代数组合的传统库之外。
原文摘要 · Abstract (English)
Discovering governing equations from data remains challenging when the underlying dynamics involve nonlocal differential operators, field interactions governed by auxiliary equations, or temporal memory effects. We propose Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO), a framework that extends Equation Learner-type symbolic architectures by incorporating pretrained neural operators as nodes in the symbolic network. NOMTO represents candidate equations as sparse differentiable computational graphs that combine algebraic operations with fixed neural operator surrogates pretrained to approximate nonlinear operators. We evaluate the method on model-discovery problems involving nonlocal spatial operators, couplings mediated by auxiliary field equations, and temporal integral terms representing memory effects. The results show that NOMTO can recover compact governing equations containing nonlocal operator terms, thereby extending symbolic model discovery beyond libraries restricted to local derivatives and point-wise algebraic combinations.
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