The world’s fastest supercomputers were designed for modeling physical phenomena, yet they still are not fast enough to robustly predict the impacts of climate change, to design controls for airplanes based on airflow or to accurately simulate a fusion reactor. All of these phenomena are modeled by partial differential equations (PDEs), the class of equations that describe everything smooth and continuous in the physical world, and the most common class of simulation problems in science and engineering. To solve these equations, we need faster simulations, but in recent years, Moore’s law has been slowing. At the same time, we’ve seen huge breakthroughs in machine learning (ML) along with faster hardware optimized for it. What does this new paradigm offer for scientific computing?

In “Learning Data Driven Discretizations for Partial Differential Equations”, published in Proceedings of the National Academy of Sciences, we explore a potential path for how ML can offer continued improvements in high-performance computing, both for solving PDEs and, more broadly, for solving hard computational problems in every area of science.

For most real-world problems, closed-form solutions to PDEs don’t exist. Instead, one must find discrete equations (“discretizations”) that a computer can solve to approximate the continuous PDE. Typical approaches to solve PDEs represent equations on a grid, e.g., using finite differences. To achieve convergence, the mesh spacing of the grid needs to be smaller than the smallest feature size of the solutions. This often isn’t feasible because of an unfortunate scaling law: achieving 10x higher resolution requires 10,000x more compute, because the grid must be scaled in four dimensions—three spatial dimensions and time. Instead, in our paper we show that ML can be used to learn better representations for PDEs on coarser grids.

Satellite photo of a hurricane, at both full resolution and simulated resolution in a state of the art weather model. Cumulus clouds (e.g., in the red circle) are responsible for heavy rainfall, but in the weather model the details are entirely blurred out. Instead, models rely on crude approximations for sub-grid physics, a key source of uncertainty in climate models. Image credit: NOAA

The challenge is to retain the accuracy of high-resolution simulations while still using the coarsest grid possible. In our work we’re able to improve upon existing schemes by replacing heuristics based on deep human insight (e.g., “solutions to a PDE should always be smooth away from discontinuities”) with optimized rules based on machine learning. The rules our ML models recover are complex, and we don’t entirely understand them, but they incorporate sophisticated physical principles like the idea of “upwinding”—to accurately model what’s coming towards you in a fluid flow, you should look upstream in the direction the wind is coming from. An example of our results on a simple model of fluid dynamics are shown below:

Simulations of Burgers’ equation, a model for shock waves in fluids, solved with either a standard finite volume method (left) or our neural network based method (right). The orange squares represent simulations with each method on low resolution grids. These points are fed back into the model at each time step, which then predicts how they should change. Blue lines show the exact simulations used for training. The neural network solution is much better, even on a 4x coarser grid, as indicated by the orange squares smoothly tracing the blue line.

Our research also illustrates a broader lesson about how to effectively combine machine learning and physics. Rather than attempting to learn physics from scratch, we combined neural networks with components from traditional simulation methods, including the known form of the equations we’re solving and finite volume methods. This means that laws such as conservation of momentum are exactly satisfied, by construction, and allows our machine learning models to focus on what they do best, learning optimal rules for interpolation in complex, high-dimensional spaces.

Next Steps
We are focused on scaling up the techniques outlined in our paper to solve larger scale simulation problems with real-world impacts, such as weather and climate prediction. We’re excited about the broad potential of blending machine learning into the complex algorithms of scientific computing.

Acknowledgments
Thanks to co-authors Yohai Bar-Sinari, Jason Hickey and Michael Brenner; and Google collaborators Peyman Milanfar, Pascal Getreur, Ignacio Garcia Dorado, Dmitrii Kochkov, Jiawei Zhuang and Anton Geraschenko.

Advances in machine learning (ML) have shown great promise for assisting in the work of healthcare professionals, such as aiding the detection of diabetic eye disease and metastatic breast cancer. Though high-performing algorithms are necessary to gain the trust and adoption of clinicians, they are not always sufficient—what information is presented to doctors and how doctors interact with that information can be crucial determinants in the utility that ML technology ultimately has for users.

The medical specialty of anatomic pathology, which is the gold standard for the diagnosis of cancer and many other diseases through microscopic analysis of tissue samples, can greatly benefit from applications of ML. Though diagnosis through pathology is traditionally done on physical microscopes, there has been a growing adoption of “digital pathology,” where high-resolution images of pathology samples can be examined on a computer. With this movement comes the potential to much more easily look up information, as is needed when pathologists tackle the diagnosis of difficult cases or rare diseases, when “general” pathologists approach specialist cases, and when trainee pathologists are learning. In these situations, a common question arises, “What is this feature that I’m seeing?” The traditional solution is for doctors to ask colleagues, or to laboriously browse reference textbooks or online resources, hoping to find an image with similar visual characteristics. The general computer vision solution to problems like this is termed content-based image retrieval (CBIR), one example of which is the “reverse image search” feature in Google Images, in which users can search for similar images by using another image as input.

Today, we are excited to share two research papers describing further progress in human-computer interaction research for similar image search in medicine. In “Similar Image Search for Histopathology: SMILY” published in Nature Partner Journal (npj) Digital Medicine, we report on our ML-based tool for reverse image search for pathology. In our second paper, “Human-Centered Tools for Coping with Imperfect Algorithms During Medical Decision-Making” (preprint available here), which received an honorable mention at the 2019 ACM CHI Conference on Human Factors in Computing Systems, we explored different modes of refinement for image-based search, and evaluated their effects on doctor interaction with SMILY.

SMILY Design
The first step in developing SMILY was to apply a deep learning model, trained using 5 billion natural, non-pathology images (e.g., dogs, trees, man-made objects, etc.), to compress images into a “summary” numerical vector, called an embedding. The network learned during the training process to distinguish similar images from dissimilar ones by computing and comparing their embeddings. This model is then used to create a database of image patches and their associated embeddings using a corpus of de-identified slides from The Cancer Genome Atlas. When a query image patch is selected in the SMILY tool, the query patch’s embedding is similarly computed and compared with the database to retrieve the image patches with the most similar embeddings.

Schematic of the steps in building the SMILY database and the process by which input image patches are used to perform the similar image search.

The tool allows a user to select a region of interest, and obtain visually-similar matches. We tested SMILY’s ability to retrieve images along a pre-specified axis of similarity (e.g. histologic feature or tumor grade), using images of tissue from the breast, colon, and prostate (3 of the most common cancer sites). We found that SMILY demonstrated promising results despite not being trained specifically on pathology images or using any labeled examples of histologic features or tumor grades.

Example of selecting a small region in a slide and using SMILY to retrieve similar images. SMILY efficiently searches a database of billions of cropped images in a few seconds. Because pathology images can be viewed at different magnifications (zoom levels), SMILY automatically searches images at the same magnification as the input image.

Refinement tools for SMILY
However, a problem emerged when we observed how pathologists interacted with SMILY. Specifically, users were trying to answer the nebulous question of “What looks similar to this image?” so that they could learn from past cases containing similar images. Yet, there was no way for the tool to understand the intent of the search: Was the user trying to find images that have a similar histologic feature, glandular morphology, overall architecture, or something else? In other words, users needed the ability to guide and refine the search results on a case-by-case basis in order to actually find what they were looking for. Furthermore, we observed that this need for iterative search refinement was rooted in how doctors often perform “iterative diagnosis”—by generating hypotheses, collecting data to test these hypotheses, exploring alternative hypotheses, and revisiting or retesting previous hypotheses in an iterative fashion. It became clear that, for SMILY to meet real user needs, it would need to support a different approach to user interaction.

Through careful human-centered research described in our second paper, we designed and augmented SMILY with a suite of interactive refinement tools that enable end-users to express what similarity means on-the-fly: 1) refine-by-region allows pathologists to crop a region of interest within the image, limiting the search to just that region; 2) refine-by-example gives users the ability to pick a subset of the search results and retrieve more results like those; and 3) refine-by-concept sliders can be used to specify that more or less of a clinical concept be present in the search results (e.g., fused glands). Rather than requiring that these concepts be built into the machine learning model, we instead developed a method that enables end-users to create new concepts post-hoc, customizing the search algorithm towards concepts they find important for each specific use case. This enables new explorations via post-hoc tools after a machine learning model has already been trained, without needing to re-train the original model for each concept or application of interest.

Through our user study with pathologists, we found that the tool-based SMILY not only increased the clinical usefulness of search results, but also significantly increased users’ trust and likelihood of adoption, compared to a conventional version of SMILY without these tools. Interestingly, these refinement tools appeared to have supported pathologists’ decision-making process in ways beyond simply performing better on similarity searches. For example, pathologists used the observed changes to their results from iterative searches as a means of progressively tracking the likelihood of a hypothesis. When search results were surprising, many re-purposed the tools to test and understand the underlying algorithm, for example, by cropping out regions they thought were interfering with the search or by adjusting the concept sliders to increase the presence of concepts they suspected were being ignored. Beyond being passive recipients of ML results, doctors were empowered with the agency to actively test hypotheses and apply their expert domain knowledge, while simultaneously leveraging the benefits of automation.

With these interactive tools enabling users to tailor each search experience to their desired intent, we are excited for SMILY’s potential to assist with searching large databases of digitized pathology images. One potential application of this technology is to index textbooks of pathology images with descriptive captions, and enable medical students or pathologists in training to search these textbooks using visual search, speeding up the educational process. Another application is for cancer researchers interested in studying the correlation of tumor morphologies with patient outcomes, to accelerate the search for similar cases. Finally, pathologists may be able to leverage tools like SMILY to locate all occurrences of a feature (e.g. signs of active cell division, or mitosis) in the same patient’s tissue sample to better understand the severity of the disease to inform cancer therapy decisions. Importantly, our findings add to the body of evidence that sophisticated machine learning algorithms need to be paired with human-centered design and interactive tooling in order to be most useful.

Acknowledgements
This work would not have been possible without Jason D. Hipp, Yun Liu, Emily Reif, Daniel Smilkov, Michael Terry, Craig H. Mermel, Martin C. Stumpe and members of Google Health and PAIR. Preprints of the two papers are available here and here.