[Paper Review] Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge

Emmanuel de Bézenac, Arthur Pajot+1 more|arXiv (Cornell University)|Nov 21, 2017

Model Reduction and Neural Networks42 references249 citations

TL;DR

The paper presents a deep learning architecture for forecasting sea surface temperature (SST) by learning a motion field guided by advection-diffusion physics, enabling end-to-end training with differentiable warping and comparisons to numerical and NN baselines.

ABSTRACT

We consider the use of Deep Learning methods for modeling complex phenomena like those occurring in natural physical processes. With the large amount of data gathered on these phenomena the data intensive paradigm could begin to challenge more traditional approaches elaborated over the years in fields like maths or physics. However, despite considerable successes in a variety of application domains, the machine learning field is not yet ready to handle the level of complexity required by such problems. Using an example application, namely Sea Surface Temperature Prediction, we show how general background knowledge gained from physics could be used as a guideline for designing efficient Deep Learning models. In order to motivate the approach and to assess its generality we demonstrate a formal link between the solution of a class of differential equations underlying a large family of physical phenomena and the proposed model. Experiments and comparison with series of baselines including a state of the art numerical approach is then provided.

Motivation & Objective

• Motivate how general physical knowledge can guide deep learning model design for complex transport phenomena.
• Propose a neural architecture that jointly estimates a motion field and warps the last input frame to forecast SST.
• Show formal connections between the model and advection-diffusion PDE solutions.
• Evaluate against numerical assimilation methods and neural baselines to assess competitiveness and efficiency.

Proposed method

• Introduce a two-component model: a convolutional-deconvolutional network (CDNN) to estimate a pixel-wise motion field and a differentiable warping module to predict the next SST image.
• Warp the last input image using a Gaussian-based kernel centered at the previous pixel position (advection-diffusion inspired) to obtain I_{t+1}.
• Train end-to-end by minimizing the discrepancy between the warped prediction and the true next image using a Charbonnier loss, with optional regularization terms on the motion field (divergence, magnitude, and gradient) in the loss.
• Provide a formal link to the advection-diffusion equation: I(x,t) = ∫ k(x-w,y) I0(y) dy with k as a Gaussian kernel, grounding the architecture in physics.
• Enable autoregressive multi-step forecasting by feeding the predicted image back into the network.
• Discuss how the approach generalizes to transport problems governed by advection-diffusion.

Experimental results

Research questions

• RQ1 Can modern deep learning techniques model complex physical transport phenomena as effectively as traditional PDE-based methods?
• RQ2 How can prior physical knowledge be incorporated into deep learning architectures to improve SST forecasting?
• RQ3 What is the relationship between the learned motion field and the underlying advection-diffusion dynamics?
• RQ4 Is a differentiable warping mechanism, grounded in physics, competitive with state-of-the-art numerical models and other NN baselines for SST prediction?

Key findings

• The proposed regularized model achieves the best mean squared error (MSE) on SST forecasting among tested models.
• Regularizing the motion field (divergence, magnitude, smoothness) improves performance compared to an unregularized version.
• The model with regularization attains an average MSE of 1.42 with 0.040 s runtime, outperforming numerical model 1.99 (4.8 s) and other NN baselines.
• Compared to ConvLSTM (MSE 5.76, 0.018 s) and ACNN (MSE 15.84, 0.54 s), the proposed method is more accurate and still fast.
• GAN-based baseline (Mathieu et al., 2015) achieves 4.73 MSE (0.096 s), but is outperformed by the regularized model.
• The approach yields interpretable motion fields that reflect SST advection dynamics.