Operator compression with deep neural networks

Area of Science:

• Computational mathematics
• Scientific computing
• Machine learning applications

Background:

• Partial differential operators with multiscale coefficients pose computational challenges.
• Existing methods compress operators to sparse surrogate models on a target scale.
• High-dimensional coefficient spaces and large scale variations complicate operator compression.

Purpose of the Study:

• To develop a neural network-based approach for compressing partial differential operators.
• To approximate the coefficient-to-surrogate map directly using neural networks.
• To accelerate online computation of surrogate models for multiscale operators.

Main Methods:

• A neural network is trained to approximate the map from operator coefficients to surrogate models.
• Local assembly structures of surrogates are emulated within the neural network architecture.
• The network is trained in an efficient offline phase.
• The framework is demonstrated on second-order elliptic heterogeneous diffusion operators.

Main Results:

• Achieved significant compression ratios for multiscale operators.
• Enabled substantially accelerated online computation of surrogate models via neural network forward passes.
• Demonstrated efficient training of moderately sized neural networks.
• The proposed method outperforms classical numerical upscaling approaches in speed.

Conclusions:

• Neural network-based approximation of the coefficient-to-surrogate map offers an efficient compression strategy for partial differential operators.
• The approach facilitates rapid online surrogate model generation and computation.
• This method holds promise for accelerating simulations involving complex multiscale phenomena.