More Details

Background

Neural Emulators are networks learned to efficienty predict physical phenomena, often associated with PDEs. In the simplest case this can be a linear advection equation, all the way to more complicated Navier-Stokes cases. If we work on Uniform Cartesian grids* (which this package assumes), one can borrow plenty of architectures from image-to-image tasks in computer vision (e.g., for segmentation). This includes:

• Standard Feedforward ConvNets
• Convolutional ResNets (He et al.)
• U-Nets (Ronneberger et al.)
• Dilated ResNets (Yu et al., Stachenfeld et al.)
• Fourier Neural Operators (Li et al.)

It is interesting to note that most of these architectures resemble classical numerical methods or at least share similarities with them. For example, ConvNets (or convolutions in general) are related to finite differences, while U-Nets resemble multigrid methods. Fourier Neural Operators are related to spectral methods. The difference is that the emulators' free parameters are found based on a (data-driven) numerical optimization not a symbolic manipulation of the differential equations.

(*) This means that we essentially have a pixel or voxel grid on which space is discretized. Hence, the space can only be the scaled unit cube $\mathrm{\Omega }=(0,L{)}^D$

Boundary Conditions

This package assumes that the boundary condition is baked into the neural emulator. Hence, most components allow setting boundary_mode which can be "dirichlet", "neumann", or "periodic". This affects what is considered a degree of freedom in the grid.

Dirichlet boundaries fully eliminate degrees of freedom on the boundary. Periodic boundaries only keep one end of the domain as a degree of freedom (This package follows the convention that the left boundary is the degree of freedom). Neumann boundaries keep both ends as degrees of freedom.

Constructors

There are two primary architectural constructors for Sequential and Hierarchical Networks that allow for composability with the PDEquinox blocks.

Sequential Constructor

The squential network constructor is defined by: * a lifting block $\mathcal{L}$ * $N$ blocks ${\left\{{\mathcal{B}}_i\right\}}_{i=1}^N$ * a projection block $\mathcal{P}$ * the hidden channels within the sequential processing * the number of blocks $N$ (one can also supply a list of hidden channels if they shall be different between blocks)

Hierarchical Constructor

The hierarchical network constructor is defined by: * a lifting block $\mathcal{L}$ * The number of levels $D$ (i.e., the number of additional hierarchies). Setting $D=0$ recovers the sequential processing. * a list of $D$ blocks ${\left\{{\mathcal{D}}_i\right\}}_{i=1}^D$ for downsampling, i.e. mapping downwards to the lower hierarchy (oftentimes this is that they halve the spatial axes while keeping the number of channels) * a list of $D$ blocks ${\left\{{\mathcal{B}}_i^l\right\}}_{i=1}^D$ for processing in the left arc (oftentimes this changes the number of channels, e.g. doubles it such that the combination of downsampling and left processing halves the spatial resolution and doubles the feature count) * a list of $D$ blocks ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^D$ for upsamping, i.e., mapping upwards to the higher hierarchy (oftentimes this doubles the spatial resolution; at the same time it halves the feature count such that we can concatenate a skip connection) * a list of $D$ blocks ${\left\{{\mathcal{B}}_i^r\right\}}_{i=1}^D$ for processing in the right arc (oftentimes this changes the number of channels, e.g. halves it such that the combination of upsampling and right processing doubles the spatial resolution and halves the feature count) * a projection block $\mathcal{P}$ * the hidden channels within the hierarchical processing (if just an integer is provided; this is assumed to be the number of hidden channels in the highest hierarchy.)

Beyond Architectural Constructors

For completion, pdequinox.arch also provides a ConvNet which is a simple feed-forward convolutional network. It also provides MLP which is a dense networks which also requires pre-defining the number of resolution points.

Related

Similar packages that provide a collection of emulator architectures are PDEBench and PDEArena. With focus on Phyiscs-informed Neural Networks and Neural Operators, there are also DeepXDE and NVIDIA Modulus.