By Mats Ehrnström
NTNU Norwegian University of Science and Technology
DIScover is a collaboration and centre proposal in mathematical sciences designed to make dynamical discovery a reliable and repeatable workflow. The aim is to go from noisy observations to interpretable equations, from uncertainty to prediction, and from theory to validation in experiments and systems in fluids, neuroscience, and neurolinguistics.
We have entered an era where measurement is relatively cheap and ubiquitous, but understanding is not. Vast datasets are routine. What is not routine is turning them into models we can trust — models that explain, quantify what they do not know, and remain stable when reality deviates from assumptions. DIScover is built around that bottleneck: bridging equations, uncertainty, and learning into interpretable dynamical models.
Why now: from prediction to understanding
Classical modelling starts from laws and ends with equations. Modern modelling often starts from data and ends with predictors. Both are powerful, and both fall short in partly predictable ways. Mechanistic PDE models are efficient and explainable, but hard to calibrate from sparse observations, and may break outside asymptotic regimes. Purely data-driven learners predict, but are opaque and sensitive to deviations from training data. A third option is at hand: models that are learnable from data yet expressed in equations, with calibrated uncertainty built in. DIScover’s scientific focus is to make that middle ground mathematically reliable.
DIScover in one sentence
DIScover will create next‑generation nonlinear models with uncertainty by uniting deterministic PDE analysis, stochastic methods, structure‑preserving numerics, and data‑driven learning. And we will validate them in an open case hub where methods are applied to real data.
Four objectives — one integrated workflow
DIScover is organised around four objectives (A–D). They are meant to operate in a flow: discover → quantify uncertainty → stabilise with structure → validate and iterate.
Objective A — Data‑driven PDE discovery: To learn governing equations and operators from spatio‑temporal data, when constitutive laws are unknown or only partially valid. The emphasis is on nonlocal structure, identifiability, and robustness under realistic sampling. A practical target is to move beyond library fitting and use of synthetic data. Real observations are incomplete, unevenly sampled. DIScover aims to develop discovery methods that remain interpretable in that reality.
Objective B — Uncertainty in nonlinear dynamics: To integrate stochastic PDEs and spatio‑temporal statistics so that predictions are given with calibrated uncertainty, and beyond classical Gaussian spatial regimes into time‑dependent nonlinear settings. The long-term ambition is an integrated toolbox where analysis, computation, and statistics develop a common language.
Objective C — Geometry and constraints: To respect symmetries, invariants, and manifold structure, so that learned dynamics remain stable and explainable, and to connect structure‑preserving numerics and learning. Long‑time prediction habitually fails under drift, as violations of conservation, symmetry, or geometric structure accumulate. Objective C treats structure both as a stabiliser and as an explanatory tool —for numerical methods and for learned representations.
Objective D — The case hub (testing against reality). This is a validation engine to probe theory on real data. The initial case hub spans problems chosen to expose different ways of modelling under uncertainty: canonical wake flows; real‑world surface flows from video data; neural attractor dynamics across wake and sleep; movement composition in neural populations; and neurolinguistic closure and compositionality. A common mathematical demand ties these cases together — to extract low‑dimensional, interpretable dynamics from high‑dimensional data, to quantify uncertainty in nonlinear evolution, and to validate by predictions that can be falsified.
The case hub is designed as small set of problems that can be dynamically expanded to new fields and cases, so that theory is continuously pushed by real challenges.
Bridge to companion post
DIScover’s vision can be realizable in miniature in small, concrete projects. In a companion post, we shall describe one such workflow in detail: the rediscovery of shallow‑water evolution equations from ordinary video recordings of soliton waves. The case is an archetype of Objective A (discovering interpretable PDEs from real data) and Objective D (validation by forward prediction on withheld experiments).
What DIScover is set up to deliver
Beyond individual papers, a research collaboration should deliver reusable components for others to pick up and trust. DIScover is engineered to deliver:
• Discovery pipelines: methods that report uncertainty, expose failure modes, and can be stress-tested by controlled corruptions of data.
• A principled uncertainty layer for nonlinear models: tools that propagate uncertainty through dynamics (rather than guessing it afterwards), combining stochastic PDE ideas with statistical calibration.
• Structure‑preserving numerics for learned dynamics: methods that use geometry, invariants, and symmetries to stabilise long‑time prediction and improve interpretability.
• Open benchmarks and shared validation protocols: so new methods can be compared on common cases, and progress can be measured against reality rather than convenience.
Training and community
DIScover is designed to train researchers who are comfortable moving between PDE analysis, stochastic modelling, numerical structure, and data‑driven learning — and who can collaborate meaningfully with experimental and applied groups. Workshops, schools, and an open case hub are part of this design.
For ECMI
Industry and society increasingly need models that are both accurate and accountable. Interpretability, uncertainty quantification, and stability are essential when decisions have cost. DIScover is an example of a collaboration that aims to develop the mathematics that makes such models possible — and to demonstrate them in case studies that can be shared, scrutinised, and transferred. When such collaborations are set in action, in small or large projects, the outcome will be methods, theory, and people trained to operate at the intersection where modern modelling is among its hardest: equations, data, and uncertainty.
References
[1] DIScover (NTNU): https://www.ntnu.edu/imf/discover
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