Machine learning with hard constraints: Neural Differential-Algebraic Equations (DAEs) as a general formalism
June 3 2025 in Differential Equations, Julia, Mathematics, Programming, Science, Scientific ML, Uncategorized | Tags: adjoint methods, differential-algebraic equations, julia, modelingtoolkit, neural dae, numerical solvers | Author: Christopher Rackauckas
We recently released a new manuscript Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints where we showed a way to develop neural networks where any arbitrary constraint function can be directly imposed throughout the evolution equation to near floating point accuracy. However, in true academic form it focuses directly on getting to the point about the architecture, but here I want to elaborate about the mathematical structures that surround the object, particularly the differential-algebraic equation (DAE), how its various formulations lead to the various architectures (such as stabilized neural ODEs), and elaborate on the other related architectures which haven’t had a paper yet but how you’d do it (and in what circumstances they would make sense).
READ MOREHow chaotic is chaos? How some AI for Science / SciML papers are overstating accuracy claims
May 26 2025 in Differential Equations, Julia, Mathematics, Programming, Science, Scientific ML | Tags: | Author: Christopher Rackauckas
Just how chaotic are chaotic systems? Many of you may have heard of “the butterfly effect” but don’t quite know the mathematics behind such systems. What I want to demonstrate is the “sensitive dependence to initial conditions” property of chaotic systems and just how sensitive these systems are. The reason this has come up is that I have seen some AI papers claiming to be able to predict the timeseries of a chaotic system (many more can be found online too, just highlighting a few random ones). What I want to bring to the forefront is an examination of what is really being claimed: just how hard is it to actually forecast a chaotic system? And if they aren’t doing that, what have they done instead?
Quick Understanding of Chaos: Sensitive Dependence and the Shadowing Lemma
First of … READ MORE
A Hands on Introduction to Applied Scientific Machine Learning / Physics-Informed Learning
May 11 2025 in Julia, Scientific ML, Uncategorized | Tags: ml, neural networks, sciml | Author: Christopher Rackauckas
Presented at JuliaEO25
This is a hands-on introduction to Scientific Machine Learning that does not assume a background in machine learning. We start scratch, showing the mathematical basis of “what is a neural network?” all the way up through adding physical intuition to the neural network and using it solve problem in epidemic outbreaks to improving sensor tracking of Formula 1 cars.
The Numerical Analysis of Differentiable Simulation: Automatic Differentiation Can Be Incorrect
April 20 2025 in Differential Equations, Julia, Mathematics, Scientific ML | Tags: | Author: Christopher Rackauckas
The Numerical Analysis of Differentiable Simulation: How Automatic Differentiation of Physics Can Give Incorrect Derivatives
Scientific machine learning (SciML) relies heavily on automatic differentiation (AD), the process of constructing gradients which include machine learning integrated into mechanistic models for the purpose of gradient-based optimization. While these differentiable programming approaches pitch an idea of “simply put the simulator into a loss function and use AD”, it turns out there are a lot more subtle details to consider in practice. In this talk we will dive into the numerical analysis of differentiable simulation and ask the question: how numerically stable and robust is AD? We will use examples from the Python-based Jax (diffrax) and PyTorch (torchdiffeq) libraries in order to demonstrate how canonical … READ MORE
JuliaSim: Building a Product which improves Open Source Sustainability
January 26 2025 in Differential Equations, HPC, Julia, Scientific ML | Tags: | Author: Christopher Rackauckas
How do you build products that support open source communities? In this non-technical talk with OpenTeams I discuss how the MIT Julia Lab, PumasAI, and JuliaHub have all been essential pillars of the julialang opensource community in its goal to achieve sustainable open science. If you’ve ever been curious about what the difference is between the Julia Lab and JuliaHub is, the evolution of these groups, and what kinds of different contributions they make to the open source community, in this talk I go through as many details as I could!
Integrating equation solvers with probabilistic programming through differentiable programming
November 24 2022 in Julia, Programming, Science, Scientific ML | Tags: differential equations, julia, probabilistic programming, scientific machine learning, sciml | Author: Christopher Rackauckas
Part of the COMPUTATIONAL ABSTRACTIONS FOR PROBABILISTIC AND DIFFERENTIABLE PROGRAMMING WORKSHOP
Abstract: Many probabilistic programming languages (PPLs) attempt to integrate with equation solvers (differential equations, nonlinear equations, partial differential equations, etc.) from the inside, i.e. the developers of the PPLs like Stan provide differential equation solver choices as part of the suite. However, as equation solvers are an entire discipline to themselves with many active development communities and subfields, this places an immense burden on PPL developers to keep up with the changing landscape of tens of thousands of independent researchers. In this talk we will explore how Julia PPLs such as Turing.jl support of equation solvers from the outside, i.e. how the tools of differentiable programming allows equation solver libraries to be compatible with … READ MORE
Direct Automatic Differentiation of (Differential Equation) Solvers vs Analytical Adjoints: Which is Better?
October 11 2022 in Differential Equations, Julia, Mathematics, Science, Scientific ML | Tags: automatic differentiation, differentiable programming, sciml | Author: Christopher Rackauckas
Automatic differentiation of a “solver” is a subject with many details for doing it in the most effective form. For this reason, there are a lot of talks and courses that go into lots of depth on the topic. I recently gave a talk on some of the latest stuff in differentiable simulation with the American Statistical Association, and have some detailed notes on such adjoint derivations as part of the 18.337 Parallel Computing and Scientific Machine Learning graduate course at MIT. And there are entire organizations like my SciML Open Source Software Organization which work day-in and day-out on the development of new differentiable solvers.
I’ll give a brief summary of all my materials here below.
Continuous vs Discrete Differentiation of Solvers
AD of a solver can be done in essentially two different ways: either directly performing automatic … READ MORE
Is Differentiable Programming Actually Necessary? Can’t you just train the neural networks separately?
October 4 2022 in Scientific ML | Tags: | Author: Christopher Rackauckas
Is differentiable programming actually necessary, or can you just train the neural network in isolation against data and then stick the trained neural network into the simulation? We looked at this problem in detail in our new manuscript titled Capturing missing physics in climate model parameterizations using neural differential equations.
The goal of this project is to understand temperature mixing in large eddy simulations, essentially columns of water in the ocean. I.e., can we take a “true” 3D Navier-Stokes and use that to build very quick and accurate models for how heat flows up and down in the water?
This isn’t a new problem: … READ MORE
Composing Modeling and Simulation with Julia (2021 Modelica Conference)
April 8 2022 in Differential Equations, Julia, Science, Scientific ML | Tags: | Author: Christopher Rackauckas
In this paper we introduce JuliaSim, a high-performance programming environment designed to blend traditional modeling and simulation with machine learning. JuliaSim can build accelerated surrogates from component-based models, such as those conforming to the FMI standard, using continuous-time echo state networks (CTESN). The foundation of this environment, ModelingToolkit.jl, is an acausal modeling language which can compose the trained surrogates as components within its staged compilation process. As a complementary factor we present the JuliaSim model library, a standard library with differential-algebraic equations and pre-trained surrogates, which can be composed using the modeling system for design, optimization, and control. We demonstrate the effectiveness of the surrogate-accelerated modeling and simulation approach on HVAC dynamics by showing that the CTESN surrogates accurately capture the dynamics of a HVAC … READ MORE
Engineering Trade-Offs in Automatic Differentiation: from TensorFlow and PyTorch to Jax and Julia
December 25 2021 in Julia, Programming, Science, Scientific ML | Tags: automatic differentiation, compilers, differentiable programming, jax, julia, machine learning, pytorch, tensorflow, XLA | Author: Christopher Rackauckas
To understand the differences between automatic differentiation libraries, let’s talk about the engineering trade-offs that were made. I would personally say that none of these libraries are “better” than another, they simply all make engineering trade-offs based on the domains and use cases they were aiming to satisfy. The easiest way to describe these trade-offs is to follow the evolution and see how each new library tweaked the trade-offs made of the previous.
Early TensorFlow used a graph building system, i.e. it required users to essentially define variables in a specific graph language separate from the host language. You had to define “TensorFlow variables” and “TensorFlow ops”, and the AD would then be performed on this static graph. Control flow constructs were limited to the constructs that could be represented statically. For example, an `ifelse` function statement is very different from … READ MORE