  In the last decade, deep learning has become a major component of artificial
intelligence. The workhorse of deep learning is the optimization of loss
functions by stochastic gradient descent (SGD). Traditionally in deep learning,
neural networks are differentiable mathematical functions, and the loss
gradients required for SGD are computed with the backpropagation algorithm.
However, the computer architectures on which these neural networks are
implemented and trained suffer from speed and energy inefficiency issues, due
to the separation of memory and processing in these architectures. To solve
these problems, the field of neuromorphic computing aims at implementing neural
networks on hardware architectures that merge memory and processing, just like
brains do. In this thesis, we argue that building large, fast and efficient
neural networks on neuromorphic architectures also requires rethinking the
algorithms to implement and train them. We present an alternative mathematical
framework, also compatible with SGD, which offers the possibility to design
neural networks in substrates that directly exploit the laws of physics. Our
framework applies to a very broad class of models, namely those whose state or
dynamics are described by variational equations. This includes physical systems
whose equilibrium state minimizes an energy function, and physical systems
whose trajectory minimizes an action functional. We present a simple procedure
to compute the loss gradients in such systems, called equilibrium propagation
(EqProp), which requires solely locally available information for each
trainable parameter. Since many models in physics and engineering can be
described by variational principles, our framework has the potential to be
applied to a broad variety of physical systems whose applications extend to
various fields of engineering, beyond neuromorphic computing.
