fol.deep_neural_networks#
The fol.deep_neural_networks module provides a comprehensive collection of deep learning
architectures and operator-learning frameworks designed for learning mappings between
function spaces, with a strong focus on physics-informed modeling, parametric systems,
and scientific machine learning.
This module implements state-of-the-art neural operators such as DeepONets and Fourier Neural Operators (FNOs), together with Conditional Neural Fields and meta-learning strategies. The provided models enable the approximation of complex solution operators and continuous fields arising in partial differential equations (PDEs), multiphysics simulations, and high-dimensional parametric problems, where classical surrogate models are often insufficient.
Key capabilities of this module include:
Learning nonlinear operators mapping input functions, boundary conditions, or parameters to solution fields.
Supporting data-driven, physics-informed, and hybrid training paradigms.
Handling explicit and implicit parametric operator learning formulations.
Neural operators based on DeepONets and Fourier Neural Operators for learning mappings between infinite-dimensional spaces.
Neural field representations using coordinate-based MLPs, including SIREN-style sinusoidal networks for high-frequency signal and geometry representation.
Conditional neural fields that model families of functions conditioned on parameters, latent variables, or context embeddings.
Autoencoding architectures for learning compact latent representations of solution manifolds and parametric fields.
Meta-learning and latent-variable methods for fast adaptation across tasks, parameters, and operating regimes.
Efficient spectral representations and convolutions for scalable, high-dimensional problems.
The module is designed to be extensible and modular, enabling seamless integration into scientific computing pipelines while leveraging modern deep learning frameworks.
Deep network base class#
Explicit parametric operator learning#
This submodule implements explicit parametric operator learning on discretized fields, where a fixed-dimensional parametric input space (for example control variables or Fourier coefficients) is mapped directly to a fixed-dimensional discretized field such as temperature or displacement.
The learning is unsupervised or physics-informed: no direct target fields are required. Instead, predicted fields are evaluated using physics-based loss functions (for example weighted residual- or energy-based formulations), with boundary conditions applied explicitly at the field level.
Implicit parametric operator learning#
This submodule implements implicit parametric operator learning using
coordinate-based neural fields. A fixed-dimensional parametric input (for example
control variables or parameterization features such as Fourier coefficients)
conditions a neural field represented by a coordinate-based MLP (the synthesizer),
with conditioning provided by a modulator network according to the coupling modes
implemented by fol.deep_neural_networks.nns.HyperNetwork.
The learning is unsupervised or physics-informed: predicted discretized fields are evaluated using physics-based loss functionals (for example residual- or energy-based formulations), and Dirichlet boundary conditions are enforced by explicitly inserting prescribed values across the batch.
Although training is typically performed on a fixed FE mesh, the coordinate-based synthesizer enables multi-resolution inference (and, in principle, multi-resolution training) by evaluating the conditioned neural field on alternative coordinate sets.
Meta-implicit parametric operator learning#
This submodule implements meta-implicit parametric operator learning, which extends implicit parametric operator learning by introducing per-sample latent adaptation. Instead of directly conditioning the neural field with parametric inputs, latent variables are optimized in an inner loop to minimize the physics-based loss, enabling fast adaptation without updating the main network weights.
The neural field is represented by a coordinate-based synthesizer MLP and is
conditioned through a modulator network using the coupling modes implemented by
fol.deep_neural_networks.nns.HyperNetwork. Training remains
unsupervised or physics-informed, with explicit boundary-condition
enforcement and support for multi-resolution inference.
Meta-alpha-meta implicit parametric operator learning#
This submodule implements meta-alpha-meta implicit parametric operator learning, which further extends meta-implicit parametric operator learning by introducing a learnable latent-step size in the inner-loop adaptation. As in the meta-implicit formulation, latent variables are optimized per sample to minimize the physics-based loss, but in this variant the magnitude of the latent update itself is learned jointly with the network parameters.
The neural field is represented by a coordinate-based synthesizer MLP and is
conditioned through a modulator network using the coupling modes implemented by
fol.deep_neural_networks.nns.HyperNetwork. The latent codes are adapted
using gradient-based updates, while a dedicated trainable step model controls the
latent update size, enabling improved robustness and adaptability across problem
instances.
Training remains unsupervised or physics-informed, with explicit enforcement of boundary conditions and preservation of the coordinate-based formulation, which allows multi-resolution inference by evaluating the conditioned neural field on alternative coordinate sets.
Fourier parametric operator learning#
This submodule implements Fourier parametric operator learning using a Fourier Neural Operator (FNO) on discretized fields. A fixed-dimensional parametric input space (for example control variables or parameterization features) is mapped to grid-aligned input channels and processed by the FNO to produce discretized field outputs such as temperature or displacement.
The FNO is not bound to a specific mesh resolution. Once trained, the learned operator can be evaluated on different grid resolutions, as long as the mesh is structured and uniform (for example square grids in 2D or cubic grids in 3D). This enables resolution-invariant inference across compatible discretizations.
The learning can be data-driven or physics-informed, depending on the chosen loss function, with boundary conditions enforced explicitly through the loss.
DeepONet parametric operator learning#
This submodule implements DeepONet-based parametric operator learning on discretized fields. A fixed-dimensional parametric input space conditions a DeepONet that is evaluated on FE mesh node coordinates to produce discretized field outputs such as temperature or displacement.
The learning can be data-driven or physics-informed, depending on the chosen loss function, with boundary conditions enforced explicitly through the loss and inference utilities.
Neural fields and hypernetworks#
This module provides building blocks for neural field models (e.g., SIREN-style MLPs and Fourier-feature MLPs) and DeepONets, and hypernetworks used to modulate or generate parameters for coordinate-based models. These components are commonly used in implicit neural representations, conditional neural fields, and meta-learning workflows.