PDE-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes that are required to accurately compute the PDE solution introduce an enormous number of parameters and require large scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time dependent PDEs, the adjoint method that is often employed to efficiently compute gradients and higher order derivatives requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing both the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for both the gradient and Hessian-vector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior to show how the DIAS method can be affordably extended to large scale problems without the need of checkpointing and large memory.

翻译：受偏微分方程约束的反问题是当今计算科学中最具挑战性和计算密集度最高的问题之一。为精确计算偏微分方程解所需的精细网格引入了大量参数，并需要大规模计算资源（如更多处理器和更大内存）才能在合理时间内求解此类系统。对于受时变偏微分方程约束的反问题，常用于高效计算梯度和高阶导数的伴随方法需要求解一个时间反向的所谓伴随偏微分方程，该方程依赖于每个时间步上的正向偏微分方程解。这要求在每一步存储高维正向解向量，此类操作会迅速耗尽可用内存资源。为缓解内存瓶颈，已有多种通过增加计算量来降低内存占用的方法被提出，包括检查点技术和压缩策略。在本研究中，我们提出一种接近理想可扩展性的自编码器压缩方法，以消除对检查点技术和大量内存存储的需求，从而同时降低求解时间和内存需求。我们将该方法与检查点技术及一种现成压缩方法，在地球尺度不适定地震反问题中进行了对比。结果验证了所提出的自编码器压缩方法在梯度及海森-向量乘积计算中均实现了预期的接近理想加速效果。为突出该方法的实用性，我们将自编码器压缩与数据驱动活性子空间先验相结合，展示了DIAS方法如何在无需检查点技术和大量内存的情况下，以可承受的成本扩展至大规模问题。