无需方程也能高效建模物理系统，低数据下表现更优。
GradINN: Gradient Informed Neural Network
- 用辅助网络引入梯度先验，约束主网络输出的梯度分布。
- 在少量数据下，对稳态与瞬态系统均优于标准NN和PINN。
- 适合缺乏解析方程、但有梯度规律的复杂工程建模任务。
我们提出梯度感知神经网络（GradINN），一种受物理信息神经网络（PINN）启发的方法，可高效近似那些基本控制方程完全未知或无法定义的物理系统，这在复杂工程问题中十分常见。GradINN利用对系统梯度的先验知识，在所有输入维度上约束预测函数的梯度。通过两个神经网络实现：一个用于建模目标函数，另一个表示先验信念（如平滑性）。定制化的损失函数使第一个网络在训练中强制满足由辅助网络推导出的梯度约束。我们在多种问题上验证了GradINN的优势，涵盖非时变系统（Friedman函数、斯托克斯流）和时变系统（洛特卡-沃尔泰拉方程、伯格斯方程）。实验结果表明，相较于标准神经网络和类似PINN的方法，其在所有测试场景中均表现出色。
原文摘要 · Abstract (English)
We propose Gradient Informed Neural Networks (GradINNs), a methodology inspired by Physics Informed Neural Networks (PINNs) that can be used to efficiently approximate a wide range of physical systems for which the underlying governing equations are completely unknown or cannot be defined, a condition that is often met in complex engineering problems. GradINNs leverage prior beliefs about a system's gradient to constrain the predicted function's gradient across all input dimensions. This is achieved using two neural networks: one modeling the target function and an auxiliary network expressing prior beliefs, e.g., smoothness. A customized loss function enables training the first network while enforcing gradient constraints derived from the auxiliary network. We demonstrate the advantages of GradINNs, particularly in low-data regimes, on diverse problems spanning non time-dependent systems (Friedman function, Stokes Flow) and time-dependent systems (Lotka-Volterra, Burger's equation). Experimental results showcase strong performance compared to standard neural networks and PINN-like approaches across all tested scenarios.
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