Contact UsIn the world of science and engineering, simulations are the engines that power discovery. Whether we’re forecasting weather, modelling fluid flow around an airplane wing, predicting heat transfer in a battery, or understanding chemical reactions, we rely on mathematical models to describe how physical systems behave. One of the most powerful, widely used tools for describing such systems are partial differential equations or PDEs.
But there’s a problem: solving PDEs is computationally expensive, especially when simulations need to run repeatedly. As engineering ambitions and research push toward increasingly complex systems – often requiring real‑time predictions – simulation speed becomes critical and is a key limiting factor.
That’s where NEC Laboratories Europe’s groundbreaking research – Active Learning for Neural PDE Solvers – steps in. A collaborative effort between ourselves, the University of Stuttgart, SimTech, IMPRS-IS and PSL University, the work represents a major milestone in reducing the time, cost and energy required to train advanced machine-learning models that solve PDEs. The result? A stepping stone towards faster scientific discovery, more efficient engineering processes, and a new era of simulation where AI doesn’t just learn quickly – it learns smartly.
Understanding PDEs: the language of physical reality
Before diving into NEC’s contribution, it’s worth taking a moment to understand what PDEs are and why they play such a central role in scientific computing. PDEs describe how physical quantities vary over space and time across fields such as heat conduction, fluid dynamics, elasticity and electromagnetics.1 They model systems with multiple interacting variables and multiple derivatives, making them powerful but computationally demanding tools.
Traditional methods like the finite element method (FEM) or finite volume method (FVM) have long been used to solve PDEs numerically, often requiring fine discretization and enormous computational cost.2 This includes complex geometry handling, high‑resolution meshes and a high number of timesteps. As simulations become more complex, this cost becomes the bottleneck limiting innovation.
The rise of neural PDE solvers
Over the last few years, machine learning has opened a new frontier in solving PDEs. Instead of relying entirely on traditional numerical solvers, researchers train neural networks to approximate the outputs of simulations directly. Supporting literature highlights methods such as Physics‑Informed Neural Networks (PINNs), DeepONets, Fourier Neural Operators (FNOs), and hybrid neural operators, all capable of approximating complex physical systems. 1,3
However, as our researchers explain, there is a major trade‑off: training a neural PDE solver requires a massive amount of high‑quality training data. And where does that data come from? The same expensive numerical solvers you’re trying to replace.
To make simulations faster, you first need to run thousands of slow simulations. NEC’s research aims to break this cycle.
The core challenge: simulation data is expensive
Training neural PDE solvers typically involves generating thousands of PDE trajectory samples – each a full simulation of a system across time and space. For complex systems like the compressible Navier–Stokes equations, simulations are extremely costly and computationally intensive, often involving multi‑dimensional grids, turbulent flow regimes and long rollout horizons.3
This means a neural PDE solver delivers real benefit only if generating the training data costs less than the savings gained from fast inference. Today, that is often not the case.
Our solution revolves around a simple but powerful idea: Instead of training on huge random datasets, what if the model could choose the most informative examples – those that teach it the most – at every step? This is known as active learning.
What active learning means for scientific computing
Active learning traditionally appears in domains like image classification and NLP but has rarely been applied systematically to PDE solvers. PDE problems are much more complex: their outputs are high‑dimensional fields rather than scalars or labels, and uncertainty estimation is challenging.
The NEC research fills this gap through AL4PDE, a modular benchmark designed specifically for active learning applied to neural PDE solvers. Supporting literature on neural operators highlights the spectral biases, failure modes and generalization challenges that active learning can help address. 1,3
AL4PDE enables algorithms to:
Query the simulator only when necessary.
Select the most informative initial conditions.
Focus on regions of high uncertainty.
Improve performance with fewer samples.
This is the first framework to evaluate active learning strategies for PDEs across multiple architectures such as U‑Net, FNO, and SineNet – and across a wide range of PDE families including Burgers’, Kuramoto–Sivashinsky, the Combined Equation and compressible Navier–Stokes systems.
What NEC and partner research found
The results of NEC and our partner’s study are impressive.
1. Up to 71% error reduction
Through uncertainty‑based and feature‑based sampling, NEC’s active learning methods reduce average error by up to 71% compared to random sampling.
2. Better worst‑case performance
In many PDE systems, the “difficult cases” – such as chaotic regions, steep gradients or turbulent transitions – define practical usefulness. Active learning systematically identifies these difficult regions and generates training data reflecting them.
3. Consistent and reusable datasets
A major barrier in scientific machine learning is that new models often require fresh data. NEC’s approach shows that active‑learning‑generated datasets are consistent across runs and beneficial even when applied to new model architectures.
4. Lower training time for equivalent quality
By focusing on informative samples, models reach high accuracy faster – even with fewer simulations.
Why PDE problems are particularly suited to active learning
Supporting literature reveals several reasons PDE‑based machine learning greatly benefits from active learning:
Neural networks show spectral bias, meaning they naturally learn smooth, low‑frequency patterns first, but struggle with high‑frequency or chaotic dynamics unless given specific examples.1
PDE systems often contain rare but important events – shock fronts, phase transitions or unstable oscillations – which uniform random sampling may miss.
High‑dimensional PDEs (especially fluid systems) have vast parameter spaces where only certain regimes produce meaningful behavior. 3
Active learning can effectively find these “needle‑in‑a‑haystack” examples.
Putting it all together: why this matters
NEC research marks a turning point. By integrating advanced AI‑driven surrogate models with high‑performance numerical methods, NEC and our partners are reducing the computational overhead, latency and parameter‑tuning burden traditionally associated with large‑scale simulation. These techniques accelerate convergence, improve model fidelity and streamline end‑to‑end scientific workflows. The result is a simulation stack that delivers higher accuracy per compute cycle, scales efficiently across heterogeneous hardware, and enables rapid iteration for complex scientific and industrial systems.
1. Simulation becomes affordable again
Heavy simulation workloads currently limit innovation across aerospace, automotive, energy, meteorology and climate science. Reducing the volume of required simulations has immediate business value.
2. Real‑time digital twins become feasible
Fast, high‑fidelity surrogate models unlock real‑time:
Monitoring
Control
Sensitivity analysis
Predictive maintenance
This is essential for Industry 4.0 applications.
3. AI models become more physically accurate
By strategically sampling complex behaviours, models become more robust and trustworthy–critical for safety‑critical systems.
4. The method scales across PDE types
AL4PDE’s broad range of PDE families and neural architectures demonstrates that the active‑learning approach is generalizable.
A future where AI and simulation work hand‑in‑hand
The NEC work does not replace traditional PDE solvers – it amplifies their efficiency. The result is a hybrid, intelligent simulation pipeline:
Traditional solvers ensure physical consistency
Neural networks provide near‑instant predictions
Active learning ensures training efficiency
This coordinated workflow represents the future of scientific computation.
Conclusion
NEC’s Active Learning for Neural PDE Solvers demonstrates how combining physics, machine learning and intelligent data selection can fundamentally reshape how we solve complex scientific problems.
By reducing simulation costs, improving data efficiency and enhancing model accuracy, NEC’s work brings us a step closer to a world where:
Engineers can run near‑instant design optimizations
Scientists can simulate complex systems in real time
Companies can build reliable digital twins without supercomputers
AI becomes a trusted partner in engineering and discovery
This research step towards a major shift in simulation technology – one where learning is not only fast, but strategically efficient.
References
1. Luo, K., Zhao, J., Wang, Y., Li, J., Wen, J., Liang, J., & Liao, S. (2025). Physics‑Informed Neural Networks for PDE Problems: A Comprehensive Review. Artificial Intelligence Review.
2. Melchers, H., Crommelin, D., Koren, B., Menkovski, V., & Sanderse, B. (2023). Comparison of Neural Closure Models for Discretised PDEs. Computers & Mathematics with Applications.
3. Takamoto, M., Praditia, T., Leiteritz, R., MacKinlay, D., Alesiani, F., Pflüger, D., & Niepert, M. (2022). PDEBench: An Extensive Benchmark for Scientific Machine Learning. NeurIPS.