Applied Mathematics & Scientific Machine Learning
Neural networks are very good at guessing. What they do not automatically have is a reason.
An analytically derived law supplies the reference points; the network learns from them, and its error feeds back to correct the model. That loop is what this track is about.
Scientific Machine Learning starts from one simple objection to ordinary deep learning: a network can fit data beautifully while obeying none of the physics behind it. The field brings the two together, the differential equations already known and the ability to learn from data, so that a model does not merely match the measurements but also makes physical sense, even when the data is scarce and noisy.
The work runs in two complementary directions. First, embedding known laws into a model, through PINNs, Neural ODEs, or architectures that preserve structure such as energy conservation. Second, the reverse: reading unknown laws straight out of data, with methods like SINDy, when a system is too tangled to derive on paper. Between the two sit the applied problems, from controlling physical systems to industrial operational data.
For students the track offers a rare combination: questions still open at an international level that can nonetheless begin as small experiments with a clear right and wrong, because many test cases have analytical solutions to check against. The entry requirements are modest, Python, calculus, and the basics of differential equations, and the results lead in two directions at once: scientific publication and industrial problems whose data people actually use.
Algorithm Playground
The topics on this roadmap click faster when you can tinker with them yourself. algplayground is an interactive space on phx.my.id for exploring these Scientific Machine Learning ideas right in the browser, a good place to start before settling on a thesis direction.
Open the Algorithm Playground ↗The Roadmap, and Where You Fit
Every large direction has a version a student can take on: a whole, realistic project that also lays one brick in a long-running research programme.
PINNs for a range of physical problems
Heat diffusion and wave phenomena. Stefan problems (moving boundaries, such as melting ice).
SINDy: recovering equations from data
Given only data, can an algorithm find the equations that govern its dynamics? This is not mere regression or a hunt for correlation, but a question of tracing the equations of motion out of the data itself.
Neural ODEs in time series
Time-series prediction that looks at the differential structure in the dynamics between variables.
Analysing industrial operational data
From anomaly detection to forecasting, a wide range of analytical tools can be brought to bear.
RL for controlling physical systems
Can a reinforcement learning (RL) agent steer a physical system, say by modulating heat flux so that a freezing front follows a target?
Publications on This Track
A selection from the full list on Google Scholar.
Financial distress prediction in emerging markets by using a machine learning approach for financial sustainability
Machine learning for predicting corporate financial distress in emerging markets, the applied side of this track, where the data is real and the results get used.
On string vibrations influenced by a smooth obstacle at one of the endpoints
String vibration against a smooth obstacle, a moving-contact problem, taken apart analytically with multiple time-scale perturbation, a class of problem usually only reachable numerically.
On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition
A closed-form solution for a melting front (a Stefan problem) whose boundary moves along with it, a correct answer against which a neural network's error can actually be measured.
On a multiple timescales perturbation approach for a Stefan problem with a time-dependent heat flux at the boundary
The heat-flux variant of the same moving-boundary problem; together with two other Q1 papers it forms an analytical benchmark most Physics-Informed Neural Network work does not have.