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## ‣ Efficient methods for solving multi-rate partial differential equations in radio frequency applications

## ‣ Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations

## ‣ Reliable Real-Time Optimization of Nonconvex Systems Described by Parametrized Partial Differential Equations

## ‣ Reduced-Basis Methods for Inverse Problems in Partial Differential Equations

## ‣ Sistema p-Fuzzy aplicado às equações diferenciais parciais; Model P-Fuzzy applied to partial differential equations

## ‣ Average and deviation for slow-fast stochastic partial differential equations

## ‣ Long-time analysis of Hamiltonian partial differential equations and their discretizations; Langzeitverhalten Hamiltonscher partieller Differentialgleichungen und ihrer Diskretisierungen

## ‣ On Symbolic Solutions of Algebraic Partial Differential Equations

## ‣ Aspects of overdetermined systems of partial differential equations in projective and conformal geometry

## ‣ Um estudo sobre o espalhamento da dengues usando equações diferenciais parciais e logica fuzzy; A study of the spread of dengue using partial differential equations and fuzzy logic

## ‣ 18.303 Linear Partial Differential Equations, Fall 2005; Linear Partial Differential Equations

## ‣ 18.303 Linear Partial Differential Equations, Fall 2004; Linear Partial Differential Equations

## ‣ Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations

## ‣ Reliable Real-Time Solution of Parametrized Elliptic Partial Differential Equations: Application to Elasticity

## ‣ Stochastic Partial Differential Equations on Evolving Surfaces and Evolving Riemannian Manifolds

## ‣ On Degenerate Partial Differential Equations

## ‣ Space-like Weingarten surfaces in the three-dimensional Minkowski space and their natural partial differential equations

## ‣ The inverse problem of the calculus of variations for systems of second-order partial differential equations in the plane

## ‣ On some partial differential equation models in socio-economic contexts - analysis and numerical simulations

## ‣ Solving Partial Differential Equations Using Artificial Neural Networks

This thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in non-stationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,

CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic

and parabolic PDEs with changing parameters and non-homogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to non-stationary environments.

The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore...