Guaranteed Approximation Bounds for Mixed-Precision Neural Operators (2307.15034v3)

Abstract: Neural operators, such as Fourier Neural Operators (FNO), form a principled approach for learning solution operators for PDEs and other mappings between function spaces. However, many real-world problems require high-resolution training data, and the training time and limited GPU memory pose big barriers. One solution is to train neural operators in mixed precision to reduce the memory requirement and increase training speed. However, existing mixed-precision training techniques are designed for standard neural networks, and we find that their direct application to FNO leads to numerical overflow and poor memory efficiency. Further, at first glance, it may appear that mixed precision in FNO will lead to drastic accuracy degradation since reducing the precision of the Fourier transform yields poor results in classical numerical solvers. We show that this is not the case; in fact, we prove that reducing the precision in FNO still guarantees a good approximation bound, when done in a targeted manner. Specifically, we build on the intuition that neural operator learning inherently induces an approximation error, arising from discretizing the infinite-dimensional ground-truth input function, implying that training in full precision is not needed. We formalize this intuition by rigorously characterizing the approximation and precision errors of FNO and bounding these errors for general input functions. We prove that the precision error is asymptotically comparable to the approximation error. Based on this, we design a simple method to optimize the memory-intensive half-precision tensor contractions by greedily finding the optimal contraction order. Through extensive experiments on different state-of-the-art neural operators, datasets, and GPUs, we demonstrate that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.

• The paper introduces a mixed-precision training method for Fourier Neural Operators that integrates tanh pre-activation to stabilize FFT computations.
• Experiments demonstrate up to a 50% reduction in memory usage and a 58% increase in training throughput on challenging PDE tasks.
• It validates the method’s effectiveness on tensorized architectures, balancing computational efficiency with maintained numerical accuracy.

Speeding up Fourier Neural Operators via Mixed Precision: A Methodological Insight

The paper "Speeding up Fourier Neural Operators via Mixed Precision" addresses a significant computational bottleneck in the training of Fourier Neural Operators (FNOs) deployed for solving high-resolution partial differential equations (PDEs). While neural operators such as FNOs have revolutionized approaches to approximate solutions for PDEs, their computational and memory demands pose challenges for practical application in large-scale systems, such as weather forecasting and climate modeling. This research focuses on leveraging mixed-precision computation to alleviate these challenges, providing substantial insights and experimental results.

Methodology and Contributions

The primary methodological advancement in this paper is the implementation of a mixed-precision training strategy for FNOs, which involves the adaptation of existing techniques to the complex-valued operations inherent in Fourier space. The core contributions of the research are detailed below:

1. Profiling of Memory and Runtime: The paper initiates with a rigorous profiling of FNO computation, quantifying the potential efficiency gains from mixed-precision arithmetic. The analysis emphasizes the computational overhead of full-precision operations in function spaces, highlighting FFTs as key targets for optimization.
2. Mixed-Precision Stabilization Techniques: Transitioning to mixed precision in FNOs introduces risks of numerical instability, such as overflow and underflow, notably during FFT operations. The authors propose the use of pre-activation functions, specifically the hyperbolic tangent (tanh), before FFT operations to mitigate these instabilities without significantly compromising accuracy. This contribution is validated by comparing against alternative stabilization methods such as hard-clipping and 2σ-clipping.
3. Efficiency and Performance Balancing: Experimentally, the authors demonstrate memory usage reductions up to 50% and improvements in training throughput by up to 58%. This is achieved with minimal loss in accuracy, established through extensive evaluations on Navier-Stokes and Darcy flow datasets. A precision scheduling mechanism further enhances performance by initially allowing for quick, approximate calculations which progressively refine over the course of training.
4. Compatibility with Tensorized FNO: The integration of these methods with the recent tensorized FNO architecture exemplifies the diversity of the operational improvements, combining reduced processing times with enhanced computational performance.

Experimental Results

The experimental analysis reveals that tanh pre-activation allows for a considerable reduction in epoch runtime, particularly when combined with Automatic Mixed Precision (AMP) tools in PyTorch, achieving a 36.8% speed increase on V100 GPUs. Additionally, the precision scheduling contributes to superior final accuracy relative to using conventional full-precision throughout training.

On a broader scale, the results subjectively underscore the trade-offs between computational efficiency and numerical precision—a balance crucial for large-scale implementations. Importantly, the zero-shot super-resolution tests illustrate the model's robustness across different resolutions, marking a key strength of neural operators in practical scenarios where training data resolutions may not perfectly match test conditions.

Implications and Future Directions

The successful deployment of mixed-precision techniques in FNOs marks a crucial step toward more resource-efficient neural operator models. The paper's findings enhance the viability of real-time applications of FNOs in domains necessitating rapid PDE solutions without prohibitive computational cost, particularly for high-resolution datasets.

Future research could explore further enhancements in mixed-precision arithmetic, such as integration with quantization techniques or synchronous distributed training environments, to push the boundaries of both computational efficiency and scalability. Furthermore, extending these techniques to other neural operator architectures, beyond FNOs, could yield broadening real-world applicability and performance improvements. Hence, this work provides both foundational methods and a stimulus for continuous innovation in neural operator research for PDE applications.