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Physically Interpretable Neural Networks for the Geosciences: Applications to Earth System Variability
Benjamin A. Toms, Elizabeth A. Barnes, Imme Ebert-Uphoff
TL;DR
Geoscientific neural networks often provide accurate outputs without revealing how they make decisions. This paper uses backwards optimization and layerwise relevance propagation to project decisions onto input dimensions, showing that interpretation can uncover physically meaningful relationships in known climate patterns. The authors argue that interpretation itself can serve as a scientific outcome, while emphasizing that network design must balance accuracy with interpretability.
Problem
Geoscientific neural networks are difficult to interpret, so their outputs are often prioritized while reasoning is used only as a secondary check.
Method
The paper applies backwards optimization and layerwise relevance propagation to trace neural-network decisions back onto original input dimensions.
Results
The methods reliably support physically meaningful inference from neural networks applied to established climate patterns, including ENSO and seasonal prediction.
Takeaways & Limitations
Interpretation can be used as the ultimate scientific outcome of a neural network and may support discovery of unknown geoscientific patterns.
Takeaways & Limitations
Backwards optimization produces a composite pattern that may combine multiple modes of variability and cannot show how those modes evolve independently or together.
Abstract
from arXiv · showhide
Neural networks have become increasingly prevalent within the geosciences, although a common limitation of their usage has been a lack of methods to interpret what the networks learn and how they make decisions. As such, neural networks have often been used within the geosciences to most accurately identify a desired output given a set of inputs, with the interpretation of what the network learns used as a secondary metric to ensure the network is making the right decision for the right reason. Neural network interpretation techniques have become more advanced in recent years, however, and we therefore propose that the ultimate objective of using a neural network can also be the interpretation of what the network has learned rather than the output itself. We show that the interpretation of neural networks can enable the discovery of scientifically meaningful connections within geoscientific data. In particular, we use two methods for neural network interpretation called backwards optimization and layerwise relevance propagation, both of which project the decision pathways of a network back onto the original input dimensions. To the best of our knowledge, LRP has not yet been applied to geoscientific research, and we believe it has great potential in this area. We show how these interpretation techniques can be used to reliably infer scientifically meaningful information from neural networks by applying them to common climate patterns. These results suggest that combining interpretable neural networks with novel scientific hypotheses will open the door to many new avenues in neural network-related geoscience research.
1 Introduction
Geoscientific neural networks are often treated as black boxes whose outputs are optimized while their reasoning remains difficult to interpret. The paper proposes using interpretation itself to infer physically meaningful relationships and test this approach on known climate variability.
- Neural networks are widely used in geoscience for prediction, pattern identification, and parameterizing sub-grid physics.
- Their increasing architectural complexity makes it difficult to deduce how trained networks combine information to produce outputs.
- Geoscientific applications commonly prioritize output accuracy, using interpretation mainly to check whether decisions align with current understanding.
- The paper proposes interpretation, rather than network output, as the ultimate scientific objective for neural-network applications.
- Backwards optimization and layerwise relevance propagation trace decisions back to input dimensions to identify important variables and physically meaningful relationships.
- The methods are applied first to established climate phenomena, including ENSO and seasonal prediction, to verify that interpretations capture known geophysical variability.
2 Neural Network Architecture
The study uses shallow, fully connected classifier networks that map vectorized geospatial images to two-class likelihoods. This basic architecture is sufficient for the examples while satisfying restrictions imposed by the interpretation methods.
- Each network has an input layer, two hidden layers with eight nodes each, and a two-node output layer for binary classification.
- The inputs are vectorized geospatial maps labeled with two-unit class vectors, while softmax outputs give class likelihoods summing to 1.
- The examples use a basic shallow architecture rather than more advanced or deeper alternatives such as convolutional neural networks.
- The basic architecture captures the examples’ known input-output relationships, while the interpretation techniques restrict permissible network structures.
3 Neural Network Interpretation Methods
The paper presents backwards optimization and layerwise relevance propagation as complementary ways to interpret trained neural networks by tracing decisions back to input patterns or features. It also identifies important scope limitations, especially for composite optimal inputs and regression-oriented LRP.
- Backwards Optimization (Optimal Input): Backwards optimization iteratively updates an input sample to maximize a trained network’s confidence in a user-specified output.The network’s weights and biases remain frozen while the input is optimized using backpropagated gradients.
- Backwards Optimization (Optimal Input): The backwards optimization procedure initializes an input, compares its output with a desired target, and repeats input updates until the error no longer decreases.The procedure uses the same loss function as network training and can initialize the sample with zeros or other choices.
- Backwards Optimization (Optimal Input): The resulting optimal input provides a composite view of patterns associated with a particular output, but may combine multiple modes of variability without showing their separate evolution.Optimizing an actual input sample rather than an all-zero sample is identified as one possible adaptation for such applications.
- Layer-Wise Relevance Propagation (LRP): LRP analyzes one input sample at a time by propagating relevance from the output layer back to the original input dimensions.For image inputs, it produces a heatmap identifying regions important for that sample’s network output.
- Layer-Wise Relevance Propagation (LRP): The LRP formulation used here propagates only information that positively contributes to categorical output confidence, ignoring negative weights, biases, and confidence-reducing information.The propagation rules conserve total summed relevance so that relevance at the input layer equals the output value.
- Layer-Wise Relevance Propagation (LRP): The authors identify limitations for regression because the formulation does not distinguish inputs that increase from those that decrease the final output.They suggest that regression-specific LRP formulations are needed, while noting that tasks can sometimes be reformulated categorically.
4 Applications to Earth System Variability
The paper applies backwards optimization and layerwise relevance propagation to interpret neural networks in two geoscientific applications: ENSO phase identification and seasonal surface-temperature prediction. The methods recover known spatial structures and reveal sample-specific and lead-time-dependent SST patterns used by the networks.
- Experimental applications: The experiments use vectorized geospatial fields to classify ENSO phases and predict the sign of surface-temperature anomalies from SST patterns.The ENSO task distinguishes El Niño from La Niña, while the seasonal task predicts temperature anomalies along the west coast of North America.
- ENSO phase identification: 100% accuracy was achieved on both training (654 samples) and testing (168 samples) datasets for ENSO phase identification.A linear regression approach reached 81.5% accuracy, indicating that nonlinearities helped describe ENSO’s spatial structure in this example.
- ENSO phase identification: Backwards optimization recovered observed ENSO-like SST anomaly patterns, while LRP highlighted the tropical Pacific and sample-specific regions relevant to eastern and central Pacific events.LRP also identified non-zero relevance in the North Pacific and refocused attention across individual El Niño cases.
- Seasonal prediction: The seasonal prediction network reached 67% accuracy for 30- to 90-day average surface-temperature anomalies, with interpretations identifying similar tropical- and North-Pacific SST regions to prior studies.At 60-, 120-, and 180-day leads, the tropical Pacific remained a predominant source of extended predictability.
- Seasonal prediction: Across prediction ranges, neural-network interpretations suggest that the North Pacific modulates concurrent west-coast temperature anomalies while the tropical Pacific provides extended lead predictability.The neural network was more accurate than regression for all prediction ranges, although the improvement was described as slight.
5 Discussion and Conclusions
Interpretable neural networks can produce physically meaningful inferences from geoscientific data, making interpretation itself a scientific objective. Backwards optimization and LRP reveal complementary patterns, but their value depends on carefully balancing network complexity and interpretability.
- The study shows that interpreting how and why a neural network makes decisions can serve as the ultimate scientific outcome.
- Backwards optimization identifies input patterns that maximize confidence for a chosen category while retaining the input’s dimensions and units.
- LRP evaluates samples individually, showing which characteristics matter for each output and how the network’s focus varies across cases.
- Scientific usefulness requires a network complex enough to capture relationships but not so complex that interpretation methods become unusable.
- Interpretation techniques can support discovery of unknown geoscientific patterns, although this application remains a subject for future research.
Appendix A Additional Neural Network Details
The appendix describes the neural network’s input connectivity, probabilistic output, loss function, optimization, regularization, and broader methodological background.
- Each vectorized grid-cell value is treated as an independent input connected to every node in the first hidden layer.
- A softmax operator converts neural-network outputs into relative class likelihoods before accuracy is estimated.
- The cross-entropy loss compares the softmax output vector with the label vector on a logarithmic scale and penalizes large errors more severely.
- Training uses gradient descent with Nesterov accelerated stochastic gradient descent for 100 epochs, reducing the learning rate by 0.5 after 50 epochs.
- L2 regularization encourages the network to distribute attention across more input nodes, with parameters differing between ENSO and seasonal-prediction problems.
- The appendix directs readers to other resources for an extended review of neural networks and their forms.