DeSyR：一种结合PINN引导结构搜索与物理信息系数优化的解耦符号恢复框架
DeSyR: A Decoupled Symbolic Recovery Framework with PINN-Guided Structure Search and Physics-Informed Coefficient Refinement
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- Chengdu University of Information Technology（成都信息工程大学）
- Sichuan University（四川大学）
- Hong Kong Polytechnic University（香港理工大学）
- Southwest Petroleum University（西南石油大学）
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中文总结 AI 辅助
DeSyR是一种结合PINN引导结构搜索与物理信息系数优化的解耦符号恢复框架，可从神经近似中恢复微分方程的紧凑显式解，在多类微分方程问题上实现了极低的优化误差与极高的收敛率。
中文摘要 AI 辅助
当不完美的教师数据指导符号拓扑搜索和系数估计时，从神经近似中恢复紧凑显式解极具挑战性。本文提出DeSyR，一种针对微分方程的解耦符号恢复框架。物理信息神经网络（PINN）指导重复搜索以构建带有临时常数的候选拓扑；拓扑确定后，仅从控制方程和规定约束中优化其系数，随后进行门控选择与验证。针对线性固定拓扑参数化，本文分析了教师误差的继承性，表明当教师误差投影到模型空间时，有限权重混合数据的物理拟合保留了O(β⁻¹)的教师相关贡献；在适定性、可表示性、零残差可达性和离散确定性条件下，纯物理优化可条件性恢复精确系数，非线性参数化则仅在局部有对应保证。DeSyR在15个微分方程问题的18种配置（涵盖高阶、时空、多维、非线性及耦合系统）上进行评估，候选级别审计显示自由参数重拟合的收敛率达99.23%，所有涉及自由系数的选定优化均收敛；配置级别中位数优化后的相对L₂误差为2.31×10⁻¹⁴或更低，在相同拓扑对比中，优化使误差降低8至14个数量级。这些结果表明，只要保留目标适配的拓扑且纯物理优化收敛，近似神经教师可指导拓扑发现，且不会将其误差尺度强加于最终恢复的系数。
英文摘要
Recovering compact explicit solutions from neural approximations is challenging when imperfect teacher data guide symbolic topology search and coefficient estimation. We present DeSyR, a decoupled symbolic recovery framework for differential equations. A physics-informed neural network guides repeated searches to construct candidate topologies with provisional constants. Once a topology is fixed, its coefficients are refined solely from the governing equation and prescribed constraints, followed by gated selection and verification. For linear fixed-topology parameterizations, we characterize teacher-error inheritance and show that finite-weight mixed data--physics fitting retains an $O(β^{-1})$ teacher-dependent contribution when the teacher error projects onto the model space. Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients; for nonlinear parameterizations, the corresponding guarantees are local. DeSyR is evaluated on 15 differential-equation problems across 18 configurations covering high-order, space--time, multidimensional, nonlinear, and coupled systems. A candidate-level audit yields a 99.23% convergence rate among free-parameter refits, while every selected refinement involving free coefficients converges. Configuration-level median refined relative $L_2$ errors are $2.31\times10^{-14}$ or lower. In same-topology comparisons, refinement reduces error by eight to fourteen orders of magnitude. These results show that an approximate neural teacher can guide topology discovery without imposing its error scale on final recovered coefficients, provided a target-capable topology is retained and physics-only refinement converges.
