UC7 — Latent Representation of PDE Solutions

Objectives

Understand why re-solving PDEs for every parameter configuration is expensive
Learn about the autoencoder-based approach to learning solution manifolds
See how cross-modal latent alignment bridges parameters and solutions
Understand how the framework handles multiple grid discretizations

Repository: wp7-UC7-latent-pde-representation Contributors: Klaus Johannsen, Yngve Heggelund (NORCE Research)

The Problem

Solving partial differential equations (PDEs) numerically is expensive. For applications that require exploring a parameterized family of solutions — varying boundary conditions, coefficients, or discretization resolutions — re-solving the PDE for every parameter configuration is prohibitive.

UC7 tackles this by learning a compact representation of the entire solution manifold, so that new solutions can be evaluated, interpolated, and compared without running the solver.

Approach: Three-Stage Latent Learning

UC7 focuses on steady-state convection–diffusion equations in two spatial dimensions, with parameterized, divergence-free convection fields.

        graph LR
    subgraph "Stage 1: Train Separately"
        A[Solution Fields] --> B[Solution<br>Autoencoder]
        C[Streamfunction<br>Parameters] --> D[Parameter<br>Autoencoder]
    end
    subgraph "Stage 2: Align"
        B --> E[Shared Latent Space]
        D --> E
    end
    subgraph "Stage 3: Fine-tune"
        E --> F[Cross-Modal<br>Reconstruction]
    end

Stage 1: Train Autoencoders Separately

Train autoencoders on two modalities independently:

Solution autoencoder — learns compact representations of PDE solution fields
Parameter autoencoder — learns compact representations of the convection streamfunction

Stage 2: Align Latent Spaces

Align the latent spaces of these different modalities so that a single shared representation bridges between parameters and solutions.

Stage 3: Fine-tune

Fine-tune encoders and decoders toward the shared latent, then evaluate reconstruction quality via relative MSE across modalities and grid discretizations.

Key Capabilities

Capability	Description
Cross-modal transfer	Map from parameter space to solution space without solving the PDE
Multi-resolution	Multiple grid discretizations coexist within the same latent space
Interpolation	Generate new solutions for parameter values not in the training set
Compression	Compact latent representations reduce storage and computation

What It Found

The framework learns structured latent spaces that capture the solution manifold:

Cross-modal alignment enables transfer between parameter space and solution space
Multiple grid discretizations coexist within the same latent space, meaning the representation works regardless of mesh resolution
The approach bridges parameters and solutions through a shared learned geometry

Educational Value

UC7 is designed as an educational sandbox for researchers interested in:

Neural operators for scientific computing
Representation learning for PDE solutions
Autoencoder architectures for scientific data
Latent space alignment across different modalities

It is the third use case (alongside UC2 and UC3) where the ML approach is shaped by the structure of the underlying mathematics, instead of treating the problem as generic regression.

Key Dependencies

Package	Role
TensorFlow	Deep learning framework (autoencoders)
NumPy	Numerical operations
SciPy	Sparse matrices, PDE solver
Matplotlib	Visualization

Quick Start

git clone https://github.com/NAICNO/wp7-UC7-latent-pde-representation.git
cd wp7-UC7-latent-pde-representation
pip install -r requirements.txt
jupyter notebook demonstrator-v1.orchestrator.ipynb

Keypoints

Re-solving PDEs for every parameter configuration is computationally prohibitive
Autoencoders learn compact representations of PDE solutions and parameters separately
Latent space alignment creates a shared representation bridging parameters and solutions
Cross-modal transfer: map from parameters to solutions without running the solver
Multiple grid discretizations coexist in the same latent space
Educational sandbox for neural operators and representation learning