Did you notice that the famous black hole image in Interstellar was born from mathematical equations? That image was possible thanks to Einstein's equations, which are a set of partial differential equations. And what's special about those equations? They describe how the world changes when it depends on several variables at the same time: space and time.
What are PDEs and why should you care?
Partial differential equations, or PDEs, are the mathematical language we use to model phenomena that depend on more than one independent variable. While an ODE describes how a single point evolves over time, a PDE gives you the full map: how something changes at every point in space and at every moment.
Think of a cake in the oven. An ODE would tell you the temperature at a single point over time. A PDE gives you the temperature map of the whole cake, with spatial coordinates and time. Sounds more complex, and it is, but it's also far more useful.
Concrete examples:
The heat equation: models diffusion and how temperature differences smooth out.
The wave equation: governs vibrations, sound and electromagnetic waves.
Navier-Stokes: describes the motion of viscous fluids; it's behind weather forecasting, aerodynamics and many medical systems. Proving existence and smoothness for it is a Millennium Prize problem by the Clay Mathematics Institute.
Financial markets: the Black–Scholes equation for option prices.
How they're solved today: classical methods and their limits
Classical numerical methods discretize the domain. For example:
Finite Differences: you approximate derivatives with differences between nearby points. Simple examples are the Euler method and Runge–Kutta schemes.
Finite Elements: you split the domain into pieces and approximate the solution with simple piecewise functions.
Finite Volume: you apply conservation laws to control volumes, very useful in fluid dynamics.
The common problem: you need a huge number of nodes for accuracy. That creates large, sparse linear systems that are expensive to solve. Also, many algorithms are inherently sequential: small time steps or neighbor dependencies limit parallelization. In practice, a small change in initial conditions forces you to repeat the whole computation.
In short: classical methods are robust but slow, costly to parallelize and hard to reuse for many different queries.
ML for PDEs: PINNs and neural operators
AI offers two main routes today:
PINNs (Physics-Informed Neural Networks): here you train a network that incorporates the PDE into the loss function. The network learns solutions that minimize the equation residual and the boundary conditions. Advantages: they're mesh-free and can incorporate physical knowledge. Limitations: hyperparameter tuning, stability and accuracy in hard cases.
Neural operators, like FNO (Fourier Neural Operator): instead of learning a solution per instance, they learn the mapping between functions (for example, from initial conditions to solution). This lets them generalize across many conditions and gives very fast inference once trained.
Why does this matter? Because these approaches let you amortize cost: you train once and perform many fast queries. They also exploit massive parallelization on GPUs, something classical methods don't fully leverage.
But it's not magic. Networks can fail in stability, may require costly synthetic data, and the community currently works with scattered benchmarks and tasks, which makes rigorous comparison difficult.
Why we need a centralized place: benchmarks and leaderboard
If you want to evaluate a model, you need clear metrics: L2 or L_inf error, conservation of physical quantities, robustness to noise, training cost, inference latency and memory use. Without standards, comparing PINNs, FNOs and numerical solvers is like comparing bicycles, motorcycles and cars without saying the slope or the distance.
Hugging Science is creating a space to centralize solvers, tasks and leaderboards. It's the same idea that drove benchmarks in NLP and vision: reproducibility, fair comparison and an entry point for researchers and developers.
How you can contribute (if you're a researcher, engineer or advanced curious)
Reproduce papers on PINNs and FNO using public datasets.
Publish checkpoints and training scripts in shared repositories.
Evaluate not only numerical error, but conservation laws and extrapolation stability.
If you work in industry: consider amortized models for real-time queries (for example, fast prediction of velocity fields in aerodynamic design). If you're a student: start with the heat or wave equation and try a simple PINN.
Final reflection
PDEs model the world around us. From visual effects to medical diagnosis and finance, understanding how to solve them fast and well has real impact. Combining classical methods with neural networks promises to speed up simulations and open new possibilities, but to move forward we need open standards and collaboration. If you're interested in building useful tools, now is a great time to join the community organizing all this.
