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{{Short description|Programming
{{Machine learning}}
'''Differentiable programming''' is a [[programming paradigm]] in which a numeric computer program can be [[Differentiation (mathematics)|differentiated]] throughout via [[automatic differentiation]].<ref name="izzo2016_dCGP">{{cite book |doi=10.1007/978-3-319-55696-3_3 |chapter=Differentiable Genetic Programming |title=Genetic Programming |series=Lecture Notes in Computer Science |date=2017 |last1=Izzo |first1=Dario |last2=Biscani |first2=Francesco |last3=Mereta |first3=Alessio |volume=10196 |pages=35–51 |arxiv=1611.04766 |isbn=978-3-319-55695-6 |s2cid=17786263 }}</ref><ref name="baydin2018automatic">{{cite journal |last1=Baydin |first1=Atilim Gunes |last2=Pearlmutter |first2=Barak A. |last3=Radul |first3=Alexey Andreyevich |last4=Siskind |first4=Jeffrey Mark |title=Automatic Differentiation in Machine Learning: a Survey |journal=Journal of Marchine Learning Research |date=2018 |volume=18 |issue=153 |pages=1–43 |url=https://jmlr.org/papers/v18/17-468.html }}</ref><ref>{{cite book |last1=Wang |first1=Fei |chapter=Backpropagation with Callbacks: Foundations for Efficient and Expressive Differentiable Programming |date=2018 |chapter-url=http://papers.nips.cc/paper/8221-backpropagation-with-callbacks-foundations-for-efficient-and-expressive-differentiable-programming.pdf |title=NIPS'18: Proceedings of the 32nd International Conference on Neural Information Processing Systems |pages=10201–10212 |last2=Decker |first2=James |last3=Wu |first3=Xilun |last4=Essertel |first4=Gregory |last5=Rompf |first5=Tiark |editor-last=Bengio |editor-first=S. |editor2-last=Wallach |editor2-first=H. |editor3-last=Larochelle |editor3-first=H. |editor4-last=Grauman |editor4-first=K |publisher=Curran Associates |ref={{harvid|NIPS'18}} }}</ref><ref name="innes">{{Cite journal|last=Innes|first=Mike|date=2018|title=On Machine Learning and Programming Languages|url=http://www.sysml.cc/doc/2018/37.pdf|journal=SysML Conference 2018|access-date=2019-07-04|archive-date=2019-07-17|archive-url=https://web.archive.org/web/20190717211700/http://www.sysml.cc/doc/2018/37.pdf|url-status=dead}}</ref><ref name="diffprog-zygote">{{cite arXiv |eprint=1907.07587 |last1=Innes |first1=Mike |last2=Edelman |first2=Alan |last3=Fischer |first3=Keno |last4=Rackauckas |first4=Chris |last5=Saba |first5=Elliot |author6=Viral B Shah |last7=Tebbutt |first7=Will |title=A Differentiable Programming System to Bridge Machine Learning and Scientific Computing |date=2019 |class=cs.PL }}</ref> This allows for [[Gradient method|gradient-based optimization]] of parameters in the program, often via [[gradient descent]], as well as other learning approaches that are based on higher-order derivative information. Differentiable programming has found use in a wide variety of areas, particularly [[scientific computing]] and [[machine learning]].<ref name="diffprog-zygote" /> One of the early proposals to adopt such a framework in a systematic fashion to improve upon learning algorithms was made by the [[Advanced Concepts Team]] at the [[European Space Agency]] in early 2016.<ref name="differential intelligence">{{Cite web |url=https://www.esa.int/gsp/ACT/projects/differential_intelligence/ |title=Differential Intelligence |date=October 2016 |access-date=2022-10-19}}</ref>
==Approaches==
Most differentiable programming frameworks work by constructing a graph containing the control flow and [[data structures]] in the program.<ref name="flux">{{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 }}</ref> Attempts generally fall into two groups:
* ''' Static, [[compiled]] graph'''-based approaches such as [[TensorFlow]],<ref group=note>TensorFlow 1 uses the static graph approach, whereas TensorFlow 2 uses the dynamic graph approach by default.</ref> [[Theano (software)|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 [[loop (computing)|loops]] or [[recursion]]), as well as making it harder for users to reason effectively about their programs.<ref name="flux" /> 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.<ref>{{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}}</ref><ref name="myia1">{{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 }}</ref><ref name="pytorchtut">{{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}}</ref>
* '''[[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 (computing)|interpreter]] overhead (particularly when composing many small operations), poorer scalability, and reduced benefit from compiler optimization.<ref name="myia1" /><ref name="pytorchtut" />
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)|Julia]] programming language{{snd}} [https://github.com/FluxML/Zygote.jl Zygote]{{snd}} works directly on Julia's [[intermediate representation]].<ref name="flux" /><ref>{{cite arXiv |eprint=1810.07951 |last1=Innes |first1=Michael |title=Don't Unroll Adjoint: Differentiating SSA-Form Programs |date=2018 |class=cs.PL }}</ref><ref name="diffprog-zygote" />
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.<ref name="flux" /><ref name="myia1" />
==Applications==
Differentiable programming has been applied in areas such as combining [[deep learning]] with [[physics engines]] in [[robotics]],<ref>{{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 }}</ref> solving [[Quantum chemistry#Electronic structure|electronic-structure]] problems with differentiable [[density functional theory]],<ref name="Li2021">{{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}}</ref> differentiable [[Ray tracing (graphics)|ray tracing]],<ref>{{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 }}</ref> [[differentiable imaging]],<ref>{{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 |hdl=10754/686576 |hdl-access=free }}</ref><ref>{{Cite journal |last=Chen |first=Ni |last2=Brady |first2=David J. |last3=Lam |first3=Edmund Y. |date=2025-07-04 |title=Differentiable Imaging: Progress, Challenges, and Outlook |url=https://spj.science.org/doi/10.34133/adi.0117 |journal=Advanced Devices & Instrumentation |language=en |volume=6 |doi=10.34133/adi.0117 |issn=2767-9713|doi-access=free }}</ref> [[image processing]],<ref>{{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 |hdl=1721.1/122623 |hdl-access=free }}</ref> and [[probabilistic programming]].<ref name="diffprog-zygote"/>
==Multidisciplinary application==
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.<ref>{{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 }}</ref>
==See also==
* [[Differentiable function]]
* [[Machine learning]]
==Notes==
{{reflist|group=note}}
==References==
{{reflist}}
{{Differentiable computing}}
{{Programming paradigms navbox}}
[[Category:Differential calculus]]
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