Laura M. Sangalli is a professor of Statistics at Politecnico di Milano, Italy. Her research interests include functional data analysis, high-dimensional and complex data, spatial data analysis, and biostatistics. With a team composed by Aldo Clemente, Alessandro Palummo and Luca Formaggia at Politecnico di Milano, and Eleonora Arnone at the University of Turin, Laura created the femR, a package for applying Finite Element Methods to solve second-order linear elliptic Partial Differential Equations over two-dimensional spaces. A grant from the R Consortium supported this project.

“femR: Finite Element Method for Solving PDEs in R” was funded by the R Consortium in 2022. What is the current status?

The femR project, generously supported last year by the R Consortium, aimed to develop a package implementing Finite Element Methods (FEM) for solving simple yet flexible forms of Partial Differential Equations (PDEs), specifically second-order linear elliptic PDEs across general two-dimensional spatial domains. This goal has been successfully achieved, and the project is now complete. However, development on the femR package continues, extending beyond the original proposal funded by the R Consortium. 

The femR package is accessible in its GitHub repository at https://github.com/fdaPDE/femR. It includes documentation and initial vignettes. Key functionalities of femR, as outlined in the proposal, are:

• Providing an interface with the RTriangle package, enabling users to build two-dimensional triangular meshes from the boundaries of spatial domains.
• Constructing finite element bases over these triangulations.
• Solving elliptic PDEs using finite element discretization.

The project’s broader objective was to introduce the R community to a foundational tool for applying Finite Element Methods to certain PDE problems. This is especially relevant considering the R community’s prevalent use of the deSolve package, which employs finite differences for discretization in differential problems.

During the development of femR, the team engaged with Karline Soetaert and Thomas Petzoldt, the authors of deSolve. These interactions led to the realization that the deSolve community had a keen interest in space-time PDE problems—a domain femR had not initially planned to address in its original proposal. Responding to this need, the team began developing capabilities within femR for solving time-dependent PDE problems, expanding the package’s scope and utility beyond its original mandate.

In summary, the femR project has not only achieved its primary goal of providing a tool for solving specific types of PDEs using FEM in the R environment but has also adapted and grown in response to community feedback and evolving scientific needs, particularly in the realm of space-time PDE problems.

What is your background, and why did you decide to propose this project?

My main research interest lies in developing Physics-Informed statistical models for spatial and functional data observed over complex multidimensional domains, such as the brain’s surface, the gray matter’s volume, a spatial region with a complex conformation, or a road network. In regularizing terms, these Physics-Informed statistical models use PDEs to embed knowledge of the Physics of the underlying phenomenon and the geometry of the domain into the statistical models. 

I love these methods since they bind together two of the richest and most powerful modeling frameworks, from statistics and mathematics. On the one hand, the regression/maximum likelihood framework for the empirical model, and on the other hand, PDEs are the most powerful mathematical tool to model complex phenomena and behaviors. 

The Physics-Informed statistical models we develop leverage on Finite Elements discretizations of the estimation problems, and they are implemented in the R package fdaPDE, which is available from CRAN for almost ten years now. This package fills a significant gap in R’s capabilities, as there was a notable absence of direct implementations of FE methods for solving PDEs, as also highlighted in the CRAN Task View on Differential Equations.

As we already discussed, deSolve enables solving some simple forms of PDEs through finite differences. However, finite differences work over regular tensorized domains (rectangular domains) and are not the natural choice to solve PDEs on more complex domains. Moreover, they cannot achieve the same level of accuracy of FEM.

Therefore, the need for an R package capable of solving PDEs using the FEM to complement deSolve. The development of femR addresses this need. It provides solutions for PDEs on complex domains and enhances accuracy, aligning perfectly with my research interests in advancing sophisticated, Physics-Informed statistical models.

Finite Element Methods (FEM) are used to solve partial differential equations (PDEs). In what situations is it used most often? Can you give some basic examples of when you would use R and FEM to solve a PDE?

Partial Differential Equations (PDEs) are extensively used in all sciences and engineering fields to model complex phenomena behaviors. However, PDEs do not usually possess closed form solutions and are typically solved numerically through discretization methods. 

The Finite Element Method stands out as one of the most popular, computationally efficient, 

and versatile approaches for discretizing and solving PDEs. (Editor’s note: In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.) This method enables the solution of PDEs on multidimensional domains with complex shapes and various boundary conditions with high accuracy.

The femR package I’ve been discussing exemplifies the application of FEM in addressing a range of partial differential problems. It can handle both spatial and spatiotemporal diffusion-transport-reaction problems. A common and straightforward example is heat conduction, modeled by the well-known heat equation. This capability finds numerous applications in various fields.

In environmental sciences, for instance, femR can be utilized to study the dispersion of pollutants released in water or air, which are transported by currents or winds. It is also adept at modeling temperature variations and pressure. In biomedicine, femR could be used to study electrophysiology and the mechanics of organs, including tumor growth dynamics.

These examples demonstrate the wide-ranging applicability of femR in solving complex PDEs across diverse scientific disciplines. The package’s ability to handle simple forms of elliptic and parabolic PDEs and more complex PDE scenarios makes it a valuable tool in the scientific community. Moving forward, there are plans to continually enhance femR, expanding its scope to address increasingly complex differential problems and applying it to additional fields of study.

Since FEM implementations were not available in R in the past, statisticians had to use external software. Now you can do this using R. Can you explain what the before and after looks like for a statistician who is using R? How big of a change is it?

Including PDEs in data analysis methods allows us to incorporate problem-specific information that may come from the problems in physics, chemistry, mechanics, or morphology of the problem into the statistical model. These physics-informed methods are gaining increasing attention from the scientific community. For example, Physics-informed Neural Networks represent a cutting-edge research direction in Artificial Intelligence. 

PDEs are not only pivotal in advanced fields like AI but are also fundamental components of epidemiological models. Additionally, they are being increasingly used in spatial data analysis and functional data analysis. These applications are particularly useful for modeling data observed over complex domains and addressing issues like anisotropy and non-stationarity.

As the complexity of data analysis challenges our community faces grows, there is an escalating need for new data analytics methods that effectively combine empirical and physical models. It’s becoming increasingly important for statisticians to be familiar with PDEs as a fundamental tool for modeling complex phenomena.

In this evolving landscape, the femR package plays a crucial role. It offers a built-in toolbox within the R environment for solving PDEs using Finite Element Methods (FEM), thus eliminating the need for users to depend on external environments. We have invested significant effort in creating an intuitive R interface. This interface simplifies formulating differential problems, making it as easy as writing them down on paper. We are confident that the availability of femR, a native and user-friendly package, will greatly aid and assist the statistical community in developing new physics-informed models.

FemR complements deSolve (https://desolve.r-forge.r-project.org/). Are there examples and documentation to help users with femR?

We have already prepared documentation and created initial vignettes for our package, available on the GitHub repository at https://fdapde.github.io/femR/.  We plan to continue enriching these vignettes with more content.

We aim to offer a tool that may complement deSolve. As mentioned earlier, we have been discussing with the authors of deSolve to understand the needs of the deSolve community. We decided to delay the release of the package on CRAN to enable a testing period in which interested users may start using the package, and we may get feedback. These are crucial as they may help us tailor the package in the most convenient way to the community. This testing period will help us avoid potential backward compatibility issues that we might otherwise encounter and ensure better integration with deSolve. 

As I mentioned, we are also actively working on incorporating other functionalities that interest the deSolve community. In particular, as part of the future development of the package, we will aim to introduce non-linear PDE terms, such as logistic terms. Indeed, adding non-linear forcing terms will enable the modeling of various biologically relevant problems, that cannot currently be tackled within femR, perhaps within the next year.

How was it working with the R Consortium? Would you recommend it to other people who work with the R programming language? Will you continue to develop femR?

We’ve had a great experience with the R Consortium, and we would recommend applying for funding to anyone who has the potential to develop packages that can benefit the R community. 

Furthermore, the grant enabled us to fund young collaborators, offering them the opportunity to gain research experience and encouraging them to pursue a Ph.D. In particular, through the grant, we supported two graduated students, Aldo Clemente and Alessandro Palummo, co-authors of femR. Aldo and Alessandro are already enrolled in a Ph.D. here at Politecnico di Milano, so we will keep working on femR and enlarge the scope of this new toolbox in several different directions toward more complex PDEs to model phenomena over different domains. We believe this may support the R community in developing new cutting-edge data analysis methods, interfacing statistical methodology and physical models. This will facilitate exploring new avenues of research.

There is a new frontier here in science! Seeds of something big! 

About ISC Funded Projects

A major goal of the R Consortium is to strengthen and improve the infrastructure supporting the R Ecosystem. We seek to accomplish this by funding projects that will improve both technical infrastructure and social infrastructure.

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