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Advanced Scientific Machine Learning
Modern Machine Learning Software
Functional Programming
A Primer on Functional Programming
Just-in-Time Compilation
Vectorization
Pseudorandom Numbers without Side Effects
Type Systems, Pytrees, and Models
Types and JAX
Python Type Annotations
Haskell Type System
Pytrees to Represent Model Parameters
Differentiable Programming
Numerical Differentiation
Symbolic Differentiation
Automatic Differentiation
Automatic Differentiation with JAX
Optimization for Scientific Machine Learning
Basics of Optimization Problems
Gradient Descent
Gradient Descent with Momentum
Optax: Optimizers in JAX
Stochastic Gradient Descent
Optimization Algorithms with Adaptive Learning Rates
Second-Order Methods for Optimization
Initialization of Neural Network Parameters
Training a Neural Network on the GPU
Uncertainty Propagation through Scientific Models
Sensitivity Analysis of ODEs
Local Sensitivity Analysis for Ordinary Differential Equations
Differentiating the Solution of Ordinary Differential Equations
Example: The Duffing Oscillator
Example: Lorenz System
Beyond Local Sensitivity Analysis: The Fokker–Planck Equation
Latin Hypercube Designs
Sobol Sequences
Variance-Based Global Sensitivity Analysis
Example: Global Sensitivity Analysis of the Duffing Oscillator
Uncertainty Propagation Using Polynomial Chaos
Required Functional Analysis
Polynomial Chaos for a Uniform Input
Symbolic Construction of Polynomial Chaos for Gaussian Random Variables
Numerical Construction of Orthonormal Polynomials
Polynomial Chaos for a Scalar ODE
Polynomial Chaos in Many Dimensions
Polynomial Chaos for Vector Dynamical Systems
Limitations of Polynomial Chaos
Surrogate Models
Basic Elements of Surrogate Modeling
Example: Neural Network Surrogate
Example: Gaussian Process Surrogate
Sparse Variational Gaussian Processes
Example: Gaussian Process Regression with Large Data Sets
Multi-Fidelity Surrogates
Multi-Fidelity Modeling
Multi-Fidelity Gaussian Process Surrogates
Active Learning
Active Learning Basics
Example: Uncertainty Sampling
Embedding Symmetries in Surrogate Models
Enforcing Symmetries in Neural Networks
Euclidean Neural Networks
High-Dimensional Uncertainty Propagation
Representing Function-Valued Uncertainty
PDE Solvers as Operators
Singular Value Decomposition
Connection Between SVD and Principal Component Analysis
The Karhunen–Loève Expansion
Example: Surrogate for the Stochastic Heat Equation
Example: Surrogate for the Stochastic Heat Equation with Principal Component Analysis
Learning Scientific Solution Operators
Learning Operators
DeepONet in JAX
Fourier Neural Operator in JAX
Inverse Problems in Deterministic Scientific Models
Basics of Inverse Problems
The Classical Formulation of Inverse Problems
Example: The Catalysis Problem Using a Classical Approach
Bayesian Formulation of Inverse Problems
The Laplace Approximation
Example: The Catalysis Problem Using the Laplace Approximation
Sampling from Posteriors
Basics of Markov Chain Monte Carlo
Random-Walk Metropolis with BlackJAX
Hamiltonian Monte Carlo with BlackJAX
No-U-Turn Sampler with BlackJAX
Variational Inference
Variational Inference Foundations
Example: The Catalysis Problem Using Variational Inference
Example: 3D Particle Position Reconstruction from Images
Hierarchical Bayesian Modeling
Hierarchical Model Structure
Population Uncertainty
Improving Posterior Geometry
Amortized Inference for Hierarchical Models
Amortized Inference for Inverse Kinematics
Deterministic, Finite-Dimensional, Dynamical Systems
ODE Inverse Problems
Structural Identifiability of a Harmonic Oscillator
Example: A Dynamical System with Multiple Observed Trajectories
PDE-Constrained Inverse Problems
PDE Inverse Problems
Inferring Thermal Conductivity
Inferring the Location of a Contaminant
Data-Driven Modeling of Dynamical Systems
Sparsity-Promoting Regularization
Sparse Identification of Nonlinear Dynamics
SINDy Example 1: Linear and Cubic Systems
SINDy Example 2: Lorenz System
Neural ODEs
Physics-Informed Neural Networks (PINNs)
Basics of Physics-Informed Neural Networks
Forward Problems with PINNs
Spectral Bias of Neural Networks
Training Pathologies and Remedies
Energy Functionals
PINNs for Parametric Studies
Solving Parametric Problems Using Physics-Informed Neural Networks
Physics-Informed Neural Operators
PINNs for Inverse Problems
Recovering an Unknown Conductivity
Bayesian PINNs for Inverse Problems
Inverse Problems in Stochastic Scientific Models
Stochastic Differential Equations
Brownian Motion and Itô Calculus
Example: Brownian Motion
Stochastic Exponential Growth
Example: Ornstein–Uhlenbeck Process
Filtering and Smoothing
Particle Filtering and Smoothing
Particle Filtering for the Duffing Oscillator
Particle Smoothing for the Duffing Oscillator
State and Parameter Inference
Expectation-Maximization for State-Space Models
Duffing Calibration with Particle Monte Carlo EM
Bayesian Inference in State-Space Models
Example: System Identification with Particle MCMC
References
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