用微分方程直接编码数据，无需近似反演，更精准高效。
Explicit and data-Efficient Encoding via Gradient Flow
- 用常微分方程定义解码器的梯度流，直接生成隐空间表示
- 在少量数据下表现更优，相比传统自编码器提升数据效率
- 适合物理科学等数据稀缺领域，可无缝融入科研工作流
自编码器通常通过编码器将数据映射到低维隐空间并由解码器重建。但在物理科学中，依赖编码器反演可能导致次优表示。本文提出一种仅使用解码器的方法，通过常微分方程（ODE）定义的梯度流直接将数据编码至隐空间，避免了近似反演。通过伴随法训练解码器，证明可省去昂贵积分计算且精度损失极小。此外，提出二阶ODE变体，近似内沃斯特加速梯度下降，实现更快收敛。针对刚性ODE问题，采用以损失最小化为导向的自适应求解器，增强鲁棒性。与传统自编码器相比，该方法实现显式编码和更优数据效率，对数据稀疏的物理科学场景尤为关键。本工作为机器学习融入科学工作流程铺平道路。
原文摘要 · Abstract (English)
The autoencoder model typically uses an encoder to map data to a lower dimensional latent space and a decoder to reconstruct it. However, relying on an encoder for inversion can lead to suboptimal representations, particularly limiting in physical sciences where precision is key. We introduce a decoder-only method using gradient flow to directly encode data into the latent space, defined by ordinary differential equations (ODEs). This approach eliminates the need for approximate encoder inversion. We train the decoder via the adjoint method and show that costly integrals can be avoided with minimal accuracy loss. Additionally, we propose a $2^{nd}$ order ODE variant, approximating Nesterov's accelerated gradient descent for faster convergence. To handle stiff ODEs, we use an adaptive solver that prioritizes loss minimization, improving robustness. Compared to traditional autoencoders, our method demonstrates explicit encoding and superior data efficiency, which is crucial for data-scarce scenarios in the physical sciences. Furthermore, this work paves the way for integrating machine learning into scientific workflows, where precise and efficient encoding is critical. \footnote{The code for this work is available at \url{https://github.com/k-flouris/gfe}.}
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