A Fresh Approach to Correcting Bias in Physical Models

Scientists often model how things work in the natural world to understand cause and effect. However, these models can be too simple. When a model does not represent reality accurately, we can see differences between what the model predicts and what we actually observe. This mismatch is known as Bias, and it can be tricky because it changes depending on how complete the model is. Traditional methods for fixing these biases, such as Bayesian methods or regressive neural networks, often miss the biases or make guesses about them. This can lead to results that are not believable.

#A New Approach

To tackle this issue, we propose a new method inspired by recent work in image processing. Instead of using the usual decoder in an autoencoder, we add in a physical model followed by a layer to correct biases. This approach allows us to fix biases while also working to understand the model better, without needing to guess too much about the nature of the biases. We tested this method with two different Physical Models: one for remote sensing that looks at how light bounces off trees, and another that studies the ground movements due to volcanic activity. Our results show that our method can be as good as or better than traditional methods, and it doesn’t require us to get rid of the biases beforehand.

#The Importance of Modeling Natural Phenomena

Understanding natural phenomena through mathematical models is a basic task in science. When we want to figure out what causes something to happen, we often use model inversion. This means we deduce the underlying causes based on what we observe. Despite its widespread use, this process has not been perfect. Past methods that tried to invert models struggled with the problem of incompleteness, meaning that the model's predictions often didn’t match real-world events.

#Recent Developments

Fortunately, two recent developments encourage a fresh look at this problem. First, new sensor technologies have given us better and larger datasets of measurements. Second, Autoencoders have been utilized to reveal simpler representations from complex data. However, while these procedures are promising, they often lack a reliable way to get results that clearly explain the physics behind the data.

Some researchers have looked at using a differentiable renderer in an autoencoder to get clearer results from pictures. Some of these methods have also been applied to specific physics problems, but it remains to be seen if they can be used more broadly. This uncertainty comes from the fact that many existing physical models are not designed in a way that fits with the needs of deep learning.

Another issue is that methods based on autoencoders usually do not address the biases found in physical models. Ideally, the outputs of a physical model should match the Observations without bias. However, in reality, many models do show systematic biases because they only partially represent a complicated physical system. Addressing these biases is crucial for accurate results.

#Our Proposed Solution

We first show how to improve a straightforward physical model with a bias correction layer that learns to transform model outputs to match observed outcomes, which can boost prediction accuracy. Next, we utilize an autoencoder-like system to invert this improved model, extracting the causes of the observed data. We applied this method to two different areas of physical science:

1. Radiative Transfer Models (RTM): These models calculate how light interacts with plant life to create images based on specific characteristics of the forests.
2. Volcanic Deformation Models: These examine how the ground shifts due to activities beneath the surface, like magma movement.

We make the complex RTM fully differentiable and stable during training, which means that it can be used more extensively in deep learning workflows.

#Learning from Different Models

We evaluate the complexity of the necessary bias corrections through two models with varying completeness. By comparing our learning outcomes against traditional methods, we demonstrate that our use of autoencoders can effectively handle both model inversion and bias correction simultaneously. This suggests a clearer path towards understanding the various physical processes involved.

#Disentangled Representation Learning

A key part of our method focuses on making sense of the results we obtain. This aspect is related to disentangled representation learning, which aims to find simple, meaningful factors from more complex data. In this area, techniques like Variational Autoencoders (VAEs) are employed. They allow different influencing factors to emerge during the learning process. In contrast, Generative Adversarial Networks (GANs) have also shown promise in creating clear representations for images. However, these models usually only provide results during the learning stage and do not have an effective mechanism for inference.

For physical models, we have an advantage, as the forward model is determined and relies on a fixed set of interpretable variables. When we invert the models, we can identify the physical factors that influence the observed data.

#Inverting Physical Models

Across various scientific areas, we have created forward numerical models based on established physical principles. The process of inverting these models has many practical applications, from healthcare to climate science to understanding the Earth's structure.

In our study, we tested our approach on two specific models:

1. Radiative Transfer Models (RTM): These models simulate how light reflects off forest canopies based on certain traits. However, discrepancies often arise due to the complexities of forest structures, affecting how accurately we can infer various variables.

2. Mogi Model: This model evaluates how the ground shifts due to a pressure source, usually a magma chamber, situated deep underground. The challenge with this model arises because the volcanic changes are typically small.

Traditional methods for inverting these models include a range of techniques such as Bayesian inference and numerical optimization, and more recently, neural network regressors have also been used.

#Using Autoencoders for Inversion

While many researchers have used autoencoders to invert physical models, the distinctive aspect of our approach is the emphasis on addressing systematic biases. Our method involves training an encoder to convert measured observations into physical variables through an autoencoder structure.

We define the process of generating observations based on our physical model while accounting for biases and noise. A standard autoencoder has an encoder and a decoder, which work together to minimize the error between the generated output and the original observations. While sophisticated network architectures can effectively minimize errors, there is no guarantee that the latent variables generated are physically interpretable.

By replacing the standard decoder with our physical model, we make sure that the encoder captures interpretable physical variables that can then be used to reconstruct the observations more accurately.

In practical terms, this means that when our physical model is ideal, the relationship between the predicted outputs and observations is straight forward. However, since real-world models are often overly rigid and too simple, we need to add a non-linear bias-correction layer that enhances the model's flexibility.

#Designing the Correction Layer

We designed the correction layer to allow for just enough complex adjustments while keeping the physical meaning of the variables intact. This allows us to correct biases without losing the essential connections between the model inputs and outputs.

#Learning the Inverse in Practice

This autoencoder setup allows us to learn the inverse function while including the bias correction layers directly into the architecture.

#Physical Models for Inversion

We focus on two physical models used in our Inversions:

1. INFORM Radiative Transfer Model (RTM): This model simulates how light interacts with trees. Traditional methods often struggle with accuracy due to model simplifications.

2. Mogi Model: This simpler model examines surface displacement caused by volcanic activity. The challenges are mainly due to the complexities of interpreting small deformation signals amidst other noise.

#Making Models Differentiable

To fit our autoencoder framework, we transform the RTM implemented in NumPy into a differentiable format in PyTorch. This transformation involves rewriting the non-differentiable aspects so they can be effectively backpropagated.

By leveraging modern technology, we reduced the time it took to convert the model code significantly. This allows our model to learn more efficiently and accurately, ultimately leading to better predictions.

#Training and Testing Datasets

We collected large datasets for both models. For the RTM, we used spectral data covering various forest types over several months. The Mogi model datasets came from GNSS (Global Navigation Satellite System) stations monitoring volcanic activity over many years.

The datasets were used to train and validate our models, ensuring that we can effectively evaluate how well our methods work in real-world conditions.

#Results: Addressing Biases

In our results, we found that applying our bias correction methods significantly improved accuracy. The biases present in the RTM were highly evident when comparing generated spectra with actual observations. Our bias correction layer corrected many discrepancies, leading to more reliable outputs.

For the Mogi model, we observed similar patterns. The bias correction improved the accuracies for vertical displacements, although the results were less straightforward than for the RTM, indicating that the simpler model posed some inherent challenges.

#Evaluation of Physical Variables

We also evaluated the physical variables we learned from our models. In the RTM, we grouped data by forest types, noticing significant trends that aligned with our expectations based on existing knowledge. Without bias correction, many variable distributions appeared implausible, but our adjustments brought these distributions back into a more believable range.

For the Mogi model, although we noted that the classical approaches required extensive data filtering and assumptions, our autoencoder method showed significant promise. It allowed for more straightforward capturing of transient signals without extra preprocessing steps.

#Conclusion and Future Directions

Our study shows a promising new method for inverting physical models while correcting for biases. By integrating these processes into a unified system, we could enhance understanding and accuracy in a variety of physical sciences. However, some limitations still exist. Our approach may not extend well to complex systems with unpredictable outcomes.

In future work, we intend to apply our technique to other types of physical models and explore how we can improve the efficiency of bias corrections. We believe that there are opportunities for more advanced methods to identify the best correction layers for different types of models. This continued work will contribute to more accurate and reliable predictions in scientific research and applications.