17 research outputs found
External optimal control of fractional parabolic PDEs
In this paper we introduce a new notion of optimal control, or source
identification in inverse, problems with fractional parabolic PDEs as
constraints. This new notion allows a source/control placement outside the
domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the
Robin cases. For the fractional elliptic PDEs this has been recently
investigated by the authors in \cite{HAntil_RKhatri_MWarma_2018a}. The need for
these novel optimal control concepts stems from the fact that the classical PDE
models only allow placing the source/control either on the boundary or in the
interior where the PDE is satisfied. However, the nonlocal behavior of the
fractional operator now allows placing the control in the exterior. We
introduce the notions of weak and very-weak solutions to the parabolic
Dirichlet problem. We present an approach on how to approximate the parabolic
Dirichlet solutions by the parabolic Robin solutions (with convergence rates).
A complete analysis for the Dirichlet and Robin optimal control problems has
been discussed. The numerical examples confirm our theoretical findings and
further illustrate the potential benefits of nonlocal models over the local
ones.Comment: arXiv admin note: text overlap with arXiv:1811.0451
Fractional Deep Neural Network via Constrained Optimization
This paper introduces a novel algorithmic framework for a deep neural network
(DNN), which in a mathematically rigorous manner, allows us to incorporate
history (or memory) into the network -- it ensures all layers are connected to
one another. This DNN, called Fractional-DNN, can be viewed as a
time-discretization of a fractional in time nonlinear ordinary differential
equation (ODE). The learning problem then is a minimization problem subject to
that fractional ODE as constraints. We emphasize that an analogy between the
existing DNN and ODEs, with standard time derivative, is well-known by now. The
focus of our work is the Fractional-DNN. Using the Lagrangian approach, we
provide a derivation of the backward propagation and the design equations. We
test our network on several datasets for classification problems.
Fractional-DNN offers various advantages over the existing DNN. The key
benefits are a significant improvement to the vanishing gradient issue due to
the memory effect, and better handling of nonsmooth data due to the network's
ability to approximate non-smooth functions
Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints
In this paper, we consider non-diffusive variational problems with mixed boundary
conditions and (distributional and weak) gradient constraints. The upper bound in the constraint
is either a function or a Borel measure, leading to the state space being a Sobolev one or the space
of functions of bounded variation. We address existence and uniqueness of the model under low
regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some
cases is posed on a non-standard space of Borel measures with square integrable divergences. We also
establish existence and uniqueness of solutions to this pre-dual problem under some assumptions.
We conclude the paper by introducing a mixed finite-element method to solve the primal-dual
system. The numerical examples confirm our theoretical findings.NSFAir Force Office of Scientific Research (AFOSR)Department of NavyFunded by Naval Postgraduate School.DMS-1818772DMS-2012391DMS-1913004FA9550-19-1-0036N00244-20-1-000
Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints
In this paper, we consider non-diffusive variational problems with mixed
boundary conditions and (distributional and weak) gradient constraints. The
upper bound in the constraint is either a function or a Borel measure, leading
to the state space being a Sobolev one or the space of functions of bounded
variation. We address existence and uniqueness of the model under low
regularity assumptions, and rigorously identify its Fenchel pre-dual problem.
The latter in some cases is posed on a non-standard space of Borel measures
with square integrable divergences. We also establish existence and uniqueness
of solutions to this pre-dual problem under some assumptions. We conclude the
paper by introducing a mixed finite-element method to solve the primal-dual
system. The numerical examples confirm our theoretical findings
Learning Control Policies of Hodgkin-Huxley Neuronal Dynamics
We present a neural network approach for closed-loop deep brain stimulation
(DBS). We cast the problem of finding an optimal neurostimulation strategy as a
control problem. In this setting, control policies aim to optimize therapeutic
outcomes by tailoring the parameters of a DBS system, typically via electrical
stimulation, in real time based on the patient's ongoing neuronal activity. We
approximate the value function offline using a neural network to enable
generating controls (stimuli) in real time via the feedback form. The neuronal
activity is characterized by a nonlinear, stiff system of differential
equations as dictated by the Hodgkin-Huxley model. Our training process
leverages the relationship between Pontryagin's maximum principle and
Hamilton-Jacobi-Bellman equations to update the value function estimates
simultaneously. Our numerical experiments illustrate the accuracy of our
approach for out-of-distribution samples and the robustness to moderate shocks
and disturbances in the system.Comment: Extended Abstract presented at Machine Learning for Health (ML4H)
symposium 2023, December 10th, 2023, New Orleans, United States, 12 page
Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference
We present two neural network approaches that approximate the solutions of
static and dynamic conditional optimal transport (COT) problems, respectively.
Both approaches enable sampling and density estimation of conditional
probability distributions, which are core tasks in Bayesian inference. Our
methods represent the target conditional distributions as transformations of a
tractable reference distribution and, therefore, fall into the framework of
measure transport. COT maps are a canonical choice within this framework, with
desirable properties such as uniqueness and monotonicity. However, the
associated COT problems are computationally challenging, even in moderate
dimensions. To improve the scalability, our numerical algorithms leverage
neural networks to parameterize COT maps. Our methods exploit the structure of
the static and dynamic formulations of the COT problem. PCP-Map models
conditional transport maps as the gradient of a partially input convex neural
network (PICNN) and uses a novel numerical implementation to increase
computational efficiency compared to state-of-the-art alternatives. COT-Flow
models conditional transports via the flow of a regularized neural ODE; it is
slower to train but offers faster sampling. We demonstrate their effectiveness
and efficiency by comparing them with state-of-the-art approaches using
benchmark datasets and Bayesian inverse problems.Comment: 25 pages, 7 tables, 8 figure