differentiable programming

Most differentiable programming frameworks work by constructing a graph containing the control flow and data structures in the program.{{cite arXiv |eprint=1811.01457 |last1=Innes |first1=Michael |last2=Saba |first2=Elliot |last3=Fischer |first3=Keno |last4=Gandhi |first4=Dhairya |author5=Marco Concetto Rudilosso |author6=Neethu Mariya Joy |last7=Karmali |first7=Tejan |last8=Pal |first8=Avik |last9=Shah |first9=Viral |title=Fashionable Modelling with Flux |date=2018 |class=cs.PL }} Attempts generally fall into two groups:

• Static, compiled graph-based approaches such as TensorFlow,TensorFlow 1 uses the static graph approach, whereas TensorFlow 2 uses the dynamic graph approach by default. Theano, and MXNet. They tend to allow for good compiler optimization and easier scaling to large systems, but their static nature limits interactivity and the types of programs that can be created easily (e.g. those involving loops or recursion), as well as making it harder for users to reason effectively about their programs. A proof-of-concept compiler toolchain called Myia uses a subset of Python as a front end and supports higher-order functions, recursion, and higher-order derivatives.{{cite book |last1=Merriënboer |first1=Bart van |last2=Breuleux |first2=Olivier |last3=Bergeron |first3=Arnaud |last4=Lamblin |first4=Pascal |chapter=Automatic differentiation in ML: where we are and where we should be going |title={{harvnb|NIPS'18}} |date=3 December 2018 |volume=31 |pages=8771–8781 |chapter-url = https://papers.nips.cc/paper/2018/hash/770f8e448d07586afbf77bb59f698587-Abstract.html}}{{Cite web |last1=Breuleux |first1=O. |last2=van Merriënboer |first2=B. |date=2017 |url=https://www.sysml.cc/doc/2018/39.pdf |title=Automatic Differentiation in Myia |access-date=2019-06-24 |archive-date=2019-06-24 |archive-url=https://web.archive.org/web/20190624180156/https://www.sysml.cc/doc/2018/39.pdf |url-status=dead }}{{Cite web|url=https://pytorch.org/tutorials/beginner/examples_autograd/tf_two_layer_net.html |title=TensorFlow: Static Graphs |work=Tutorials: Learning PyTorch |publisher=PyTorch.org |access-date=2019-03-04}}
• Operator overloading, dynamic graph-based approaches such as PyTorch, NumPy's autograd package, and [https://darioizzo.github.io/audi/ Pyaudi]. Their dynamic and interactive nature lets most programs be written and reasoned about more easily. However, they lead to interpreter overhead (particularly when composing many small operations), poorer scalability, and reduced benefit from compiler optimization.

The use of just-in-time compilation has emerged recently{{when|date=April 2025}} as a possible solution to overcome some of the bottlenecks of interpreted languages. The C++ [https://bluescarni.github.io/heyoka/index.html heyoka] and Python package [https://bluescarni.github.io/heyoka.py/index.html heyoka.py] make large use of this technique to offer advanced differentiable programming capabilities (also at high orders). A package for the Julia programming language{{snd}} [https://github.com/FluxML/Zygote.jl Zygote]{{snd}} works directly on Julia's intermediate representation.{{cite arXiv |eprint=1810.07951 |last1=Innes |first1=Michael |title=Don't Unroll Adjoint: Differentiating SSA-Form Programs |date=2018 |class=cs.PL }}

A limitation of earlier approaches is that they are only able to differentiate code written in a suitable manner for the framework, limiting their interoperability with other programs. Newer approaches resolve this issue by constructing the graph from the language's syntax or IR, allowing arbitrary code to be differentiated.

Differentiable programming has been applied in areas such as combining deep learning with physics engines in robotics,{{cite arXiv |eprint=1611.01652 |last1=Degrave |first1=Jonas |last2=Hermans |first2=Michiel |last3=Dambre |first3=Joni |last4=wyffels |first4=Francis |title=A Differentiable Physics Engine for Deep Learning in Robotics |date=2016 |class=cs.NE }} solving electronic-structure problems with differentiable density functional theory,{{cite journal |last1=Li |first1=Li |last2=Hoyer |first2=Stephan |last3=Pederson |first3=Ryan |last4=Sun |first4=Ruoxi |last5=Cubuk |first5=Ekin D. |last6=Riley |first6=Patrick |last7=Burke |first7=Kieron |year=2021 |title=Kohn-Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics |journal=Physical Review Letters |volume=126 |issue=3 |pages=036401 |arxiv=2009.08551 |bibcode=2021PhRvL.126c6401L |doi=10.1103/PhysRevLett.126.036401 |pmid=33543980 |doi-access=free}} differentiable ray tracing,{{cite journal |first1=Tzu-Mao |last1=Li |first2=Miika |last2=Aittala |first3=Frédo |last3=Durand |first4=Jaakko |last4=Lehtinen |title=Differentiable Monte Carlo Ray Tracing through Edge Sampling |journal=ACM Transactions on Graphics |volume=37 |issue=6 |pages=222:1–11 |date=2018 |doi=10.1145/3272127.3275109 |s2cid=52839714 |url=https://people.csail.mit.edu/tzumao/diffrt/|doi-access=free }} differentiable imaging,{{Cite journal |last=Chen |first=Ni |last2=Cao |first2=Liangcai |last3=Poon |first3=Ting‐Chung |last4=Lee |first4=Byoungho |last5=Lam |first5=Edmund Y. |title=Differentiable Imaging: A New Tool for Computational Optical Imaging |url=https://onlinelibrary.wiley.com/doi/10.1002/apxr.202200118 |journal=Advanced Physics Research |language=en |volume=2 |issue=6 |doi=10.1002/apxr.202200118 |issn=2751-1200|doi-access=free }} image processing,{{cite journal |first1=Tzu-Mao |last1=Li |first2=Michaël |last2=Gharbi |first3=Andrew |last3=Adams |first4=Frédo |last4=Durand |first5=Jonathan |last5=Ragan-Kelley |title=Differentiable Programming for Image Processing and Deep Learning in Halide |journal=ACM Transactions on Graphics |volume=37 |issue=4 |pages=139:1–13 |date=August 2018 |doi=10.1145/3197517.3201383 |s2cid=46927588 |url=https://cseweb.ucsd.edu/~tzli/gradient_halide |doi-access=free }} and probabilistic programming.

Differentiable programming is making significant strides in various fields beyond its traditional applications. In healthcare and life sciences, for example, it is being used for deep learning in biophysics-based modelling of molecular mechanisms, in areas such as protein structure prediction and drug discovery. These applications demonstrate the potential of differentiable programming in contributing to significant advancements in understanding complex biological systems and improving healthcare solutions.{{cite journal |last1=AlQuraishi |first1=Mohammed |last2=Sorger |first2=Peter K. |title=Differentiable biology: using deep learning for biophysics-based and data-driven modeling of molecular mechanisms |journal=Nature Methods |date=October 2021 |volume=18 |issue=10 |pages=1169–1180 |doi=10.1038/s41592-021-01283-4 |pmid=34608321 |pmc=8793939 }}

• Differentiable function
• Machine learning

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