Picture an aircraft wing slicing through air. At the metal surface, friction slows the air almost to a standstill. At a short distance, the air is moving at full speed. In between is a thin, restless layer where transition happens. Engineers call it the boundary layer. Its behaviour affects drag, heat transfer and the safety of a wing at high speed.The mathematics is awkward. The governing equations are non-linear, so engineers have to build an approximation rather than calculate an answer.Ziya Uddin, who develops computational methods at BML Munjal University’s School of Engineering and Technology in Gurugram, has proposed a third way — the Physics-Informed Optimal Homotopy Analysis Method, or PI-OHAM.Ziya Uddin combines a traditional mathematical method with optimisation inspired by physics-informed machine learning.There is an important caveat. PI-OHAM has so far been demonstrated on exactly one problem — the Blasius equation, a classic boundary-layer problem.The old wayThe story begins in 1992, when Shijun Liao introduced the Homotopy Analysis Method, or HAM.The idea is to build an answer in stages: Start with a rough approximation that obeys the physical constraints, then add correction terms that bring it closer to the full solution. How quickly those corrections converge is controlled by a mathematical knob.A 2008 variant, the Optimal Homotopy Asymptotic Method, or OHAM, automated that choice. But it generally optimises against a simplified representation of the physics rather than the full equation.Physics-Informed Neural Networks, or PINNs, took the opposite approach. Instead of constructing a solution term by term, a neural network is trained to produce an answer that obeys the governing equation and boundary conditions.Ziya Uddin has worked with PINNs. But he sees two problems: They are computationally expensive and difficult to interpret. “They are a black box kind of thing,” he says. “They have no mathematical interpretation.”Splitting differencePI-OHAM keeps HAM’s mathematical structure but changes how its parameters are chosen. Instead of a human eyeballing a graph or OHAM optimising against a simplified model, PI-OHAM defines a loss function by measuring how badly the approximation violates the full problem: The governing equation, boundary conditions and available data.An automated search then adjusts and a second parameter controlling the initial approximation until that loss is minimised. The computer does the search. The mathematics remains visible. Ziya Uddin describes traditional mathematical methods as “very time-consuming, but predictive”. In his tests, he says PI-OHAM was 50-60 times faster than a standard PINN.The interpretability claim rests on HAM’s construction. The solution starts from the linear portion of the equation and builds successive corrections, leaving a mathematical structure that can be followed from one step to the next. The optimiser chooses the parameters; it does not replace that structure with an opaque learned model.That is also why PI-OHAM is not simply OHAM with more computing power. OHAM was searching against a simplified target. PI-OHAM searches against the full problem.The test case is the Blasius equation, first formulated in 1908. Its answer is already known; the reference value for the wall shear parameter is 0.332057.According to the paper, PI-OHAM reaches roughly one-part-in-a-thousand accuracy in under 50 seconds. Classical HAM takes more than 3,000 seconds to reach a comparable order of accuracy.The accuracy story is more complicated. At the highest tested order, classical HAM actually gets closer to the benchmark than PI-OHAM. It simply takes far longer — more than 3,000 seconds against about 20.The honest trade is speed for a small amount of precision, not speed at no cost.Against a PINN, Uddin reports comparable accuracy, roughly 57 times faster.For problems without a known answer, Ziya Uddin starts with something that can be checked. “We always try to take the approximate portion of the whole model,” he says, testing new effects against an established limiting case.Why it mattersThe problems PI-OHAM addresses are areas in which India is investing heavily.Solar thermal systems, collectors and storage involve non-linear heat-transfer problems. Electronics and data centres increasingly have to move enormous amounts of heat through constrained spaces. Hypersonic vehicles and atmospheric re-entry push boundary-layer calculations into much harsher territory. Process industries such as glass and paper manufacturing contain related fluid-flow problems.None of these applications are demonstrated in the paper. PI-OHAM has not solved an Indian industrial problem. It has offered a potentially faster way of solving a class of equations that turns up in several industries India is trying to expand.For now, the fairest description is simple. An old mathematical method has acquired a new optimisation trick. On its first test, the trick looks promising.Published on August 24, 2026