Archive · 24 Aug 2026
New optimisation trick speeds boundary-layer mathematics, with trade-offs
Air moving over a wing forms a thin layer that affects drag and safety.
Scientists use difficult equations to understand this layer.
Ziya Uddin created a method called PI-OHAM to solve these equations more quickly.
It combines an older mathematical technique with computer-based optimisation.
The computer chooses settings that make the proposed answer obey the full physics problem.
Unlike some neural networks, the steps of the answer can still be inspected mathematically.
In a test involving the Blasius equation, PI-OHAM was much faster than two comparison methods.
It was about as accurate as a physics-informed neural network in the reported comparison.
However, the older HAM method was slightly more accurate in the most demanding test, so the new method is faster but not always more precise.
Ziya Uddin developed PI-OHAM, combining Homotopy Analysis Method with physics-informed optimisation.
PI-OHAM selects parameters by minimising violations of the full equation, boundary conditions and available data.
On the Blasius equation, PI-OHAM reached roughly one-part-in-a-thousand accuracy in under 50 seconds.
The method was reported as about 57 times faster than a PINN and far faster than classical HAM.
Classical HAM achieved greater accuracy at the highest tested order, showing that PI-OHAM trades some precision for speed.
- Who
- Ziya Uddin of BML Munjal University’s School of Engineering and Technology developed PI-OHAM.
- What
- A new optimisation-based version of the Homotopy Analysis Method was proposed and tested on the Blasius equation.
- When
- The article was published on August 24, 2026; the source does not specify when the method was developed or tested.
- Where
- BML Munjal University’s School of Engineering and Technology in Gurugram, India.
- Why
- The method aims to solve nonlinear boundary-layer and related engineering equations faster while retaining a mathematically interpretable structure.
Speed and interpretability
Precision and validation limits
Performance versus accuracy
Speed and interpretability
PI-OHAM is reported to provide roughly one-part-in-a-thousand accuracy much faster than classical HAM and at comparable accuracy to a PINN.
Precision and validation limits
At the highest tested order, classical HAM came closer to the benchmark than PI-OHAM, although it took much longer.
Mathematical transparency versus broader testing
Speed and interpretability
PI-OHAM retains HAM’s visible, step-by-step mathematical construction rather than replacing it with an opaque neural-network model.
Precision and validation limits
The method has been demonstrated on only the Blasius equation, whose answer is already known, and its proposed industrial applications have not yet been tested.
- New method
- Physics-Informed Optimal Homotopy Analysis Method, or PI-OHAM
- Test problem
- The Blasius equation, a classic boundary-layer problem
- Reference value
- The wall shear parameter benchmark is 0.332057
- Reported accuracy
- PI-OHAM reached roughly one-part-in-a-thousand accuracy in under 50 seconds
- Speed comparison
- It was reported as roughly 57 times faster than a physics-informed neural network
- HAM comparison
- Classical HAM took more than 3,000 seconds for comparable accuracy, while PI-OHAM took about 20 seconds in the cited high-order comparison
- Current limitation
- The method has so far been demonstrated on only one problem and has not been applied to an Indian industrial problem