FERROMOTION · RUST → WASM

Which Failure Do You Have?

Hard boundary constraints just fixed one benchmark row by a large factor. You now hit a DIFFERENT row that will not converge, advection at beta = 30. Same idea, same care. What do hard constraints do here?

When a physics-informed net will not converge, decide WHICH of three failures you have before choosing a fix. One cheap experiment (fitting your own ansatz by regression, with no PDE in the loss) separates 'it cannot represent the answer' from 'it cannot find it', and a second separates the two ways of not finding it. Finish able to say why the fix that worked last time is the wrong one this time.

01Challenge

Your PINN's loss goes down and its answer is wrong. You now reach for something: Fourier features, a loss balancer, L-BFGS, hard boundary conditions, time marching. Every one of them fixes a REAL failure, and at most one of them is yours. Guessing costs a day per guess, and the loss curve will not tell you which: it looks the same in all three cases. There are only three ways a physics-informed net fails. (1) The ansatz cannot express the answer at all. (2) It can, but the boundary penalty and the residual are fighting, so the optimiser trades one against the other. (3) It can, nothing is fighting, and minimising the residual still does not lead to the solution. This lesson is a decision procedure that tells you which, in two experiments, before you spend the day.

Predict firstHard boundary constraints just fixed one benchmark row by a large factor. You now hit a DIFFERENT row that will not converge, advection at beta = 30. Same idea, same care. What do hard constraints do here?

02Model

Start by taking the PDE away. Fit your ansatz to the answer by plain regression, supervised, no residual, no boundary term, nothing physics-informed about it. You are asking one question only: CAN this network, with this hard-constraint wrapper, express this solution? If it cannot, no optimiser will rescue you and every technique on the list is beside the point; you need capacity, features, or resolution. If it CAN, you have just proved the failure is downstream of representation, and the rest of the search is about finding, not expressing. Now separate the two ways of not finding it. Impose the boundary condition STRUCTURALLY instead of by penalty, write u = A(x) + B(x)·M(x) with B vanishing where the condition lives, so there is no boundary term left to weight. If that helps, the fight was the problem. If it does not help, or makes things worse, the residual landscape is the problem, and the fix is to shorten the horizon: march in time, window by window.

03Guided practice

Worked on the benchmark's own rows, with the numbers this stack measured. STEP A, burgers, the obvious suspect: at nu = 0.01/pi the solution steepens into a shock, and the reference jumps 0.615 between neighbouring x at t = 1, so spectral bias looks certain. Take the PDE away and fit the hard-constrained ansatz to the Cole-Hopf reference by regression: rel-L2 0.0050. A much larger net reaches 0.0056, no better. Representation is NOT the limit, and the table's 0.2079 is forty times worse than what the same ansatz can express. STEP B, is it the fight? Same net, same 4096 points, same steps, same seed, changing only how the conditions are imposed: soft penalty 0.1980, hard constraint 0.3957. Hard constraints made it WORSE, so nothing was fighting. STEP C, by elimination the residual landscape is the limit, which predicts that shortening the horizon is the fix: eight time windows, each carrying its start state structurally, reach 0.0142, fourteen times the table's best, and within three times the 0.0050 the ansatz can express at all. The same procedure on advection: regression 0.0142, soft 0.8098, hard 0.9781 (worse), marched with structural start states 0.0251. The same procedure on helmholtz gives the OTHER answer: hard 0.0091 against soft 0.0244 on matched arms, so there the fight was real.

04Feedback

PASS WHEN PASS when you can name the two experiments in order and say what each one rules out: regression-without-the-PDE rules representation in or out, and soft-versus-hard on matched arms rules the boundary fight in or out. What is left is the residual landscape.

If your run fails, check:

The order matters. Run the regression fit FIRST: it is the cheapest of the three and it is the only one that can rule out a whole class of fixes.
Matched arms or nothing: same net, same points, same steps, same seed. If two things changed, you have learned nothing about either.
A bigger net that does not help is evidence, not a dead end. It is how you know capacity was never the problem.

05Retrieve & space

You fit your ansatz to the reference by plain regression and it cannot get below 20% error. Which fixes on the list are now ruled OUT, and which is left?
Hard boundary constraints fixed one row by a large factor and made two others worse. What does that tell you about what hard constraints actually do?
Why must the soft-versus-hard comparison hold the net, the points, the steps and the seed fixed, what goes wrong in the conclusion if the net differs too?

06Mastery & project

You now have a decision procedure instead of a list of techniques. Two experiments, run in order, tell you which of three failures you have, and each failure has its own fix: capacity or features for representation, structural conditions for the boundary fight, time marching for the residual landscape. The procedure is worth more than any of the fixes, because it is the part that transfers to the equation nobody has written a paper about yet.

Take a PDE of your own that a PINN handles badly. Before changing anything, run experiment 1: fit your ansatz to a reference (analytic, or a fine finite-difference solve) with the PDE removed. Write down the number BEFORE you look at it. Then run experiment 2 on matched arms. Report which of the three failures you have and what the procedure ruled out, not just what finally worked.