用可复用的反向模型替代传统迭代求解，显著降低计算成本。
Local Inverse Geometry Can Be Amortized
- 学习一个双向代理模型，将局部逆几何预处理融入反向传播。
- 在7个偏微分方程逆问题上，成功率94.8%，推理成本降低77倍。
- 适合需要高效逆问题求解的科学计算与物理建模场景。
非线性逆问题常在廉价但脆弱的一阶更新与曲率感知方法（如高斯-牛顿和莱文伯格-马夸尔特）之间权衡，后者通过反复求解基于雅可比的线性化系统获得更优方向。本文提出一种学习替代方案：将局部逆几何缓存为可复用的反向算子。框架学习一个双向代理模型 Deceptron，并通过 D-IPG（Deceptron 反向预条件梯度法）部署，该方法将残差校正后的测量空间提案回推至隐空间。核心机制是雅可比组合惩罚（JCP），训练反向雅可比作为前向雅可比的局部左逆；其运行时版本 RJCP 沿优化轨迹衡量相同的逆一致性误差。我们证明，在局部伪逆一致性下，D-IPG 与阻尼高斯-牛顿一阶等价，偏差由组合误差和条件数控制。在七个偏微分方程逆问题基准上，D-IPG 超越标准基线，六问题可靠性套件平均成功率达94.8%，主基准上推理求解成本最低降低77倍，恢复质量相当或更优。
原文摘要 · Abstract (English)
Nonlinear inverse problems often trade inexpensive but fragile first-order updates against curvature-aware methods such as Gauss-Newton and Levenberg-Marquardt, which obtain stronger directions by repeatedly solving Jacobian-based linearized systems. We propose a learned alternative: amortize local inverse geometry into a reusable reverse operator. Our framework learns a bidirectional surrogate, Deceptron, and deploys it through D-IPG (Deceptron Inverse-Preconditioned Gradient), an iterative solver that pulls residual-corrected measurement-space proposals back to latent space. The key mechanism is a Jacobian Composition Penalty (JCP), which trains the reverse Jacobian to act as a local left inverse of the forward Jacobian; its runtime counterpart, RJCP, measures the same inverse-consistency error along optimization trajectories. We prove that D-IPG is first-order equivalent to damped Gauss-Newton under local pseudoinverse consistency, with deviation controlled by composition error and conditioning. Across seven PDE inverse-problem benchmarks, D-IPG outperforms standard baselines, achieves 94.8% mean success across the six-problem reliability suite, and reaches comparable or better recovery quality at up to 77x lower inference-time solve cost on the main benchmarks.
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