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What should you check before changing the optimizer?
Confirm that the model is solving the problem you intend it to solve. Review the PDE residual, initial and boundary conditions, units, nondimensionalization, geometry, and derivative calculations. A mistake in any of these can look like an optimization failure, but identifying one requires inspecting the actual formulation and implementation.
Write down the objective in its component form. For example, a PINN may minimize a weighted sum such as L = λPDELPDE + λBCLBC + λICLIC. The terms and weights depend on the problem; the point is to make clear which constraints are being combined into the reported total.
How can you tell what is failing?
Track each loss term separately
Plot PDE, boundary-condition and initial-condition losses against training iteration, rather than relying on their sum. If total loss falls while one component remains high or rises, the aggregate is masking a constraint the model has not learned. Check the corresponding weights and, where your framework allows it, compare the gradients each term contributes to shared model parameters.
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Wang, Teng and Perdikaris identify numerical stiffness and imbalanced back-propagated gradients as a fundamental PINN failure mode. They write: “We also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training.” Their 2021 SIAM Journal on Scientific Computing paper proposes gradient-statistics-based learning-rate annealing and a more resilient architecture. It reports a 50–100× predictive-accuracy improvement across a range of computational-physics problems; that figure is not a Navier–Stokes-specific result or a prediction for an individual model.
Map residuals across space and time
Evaluate the PDE residual on a dense diagnostic grid covering the full spatial and temporal domain, not just the collocation points used for training. Look for localized peaks, broad regions of high error, or a residual distribution dominated by a small part of the flow. A low mean can conceal these patterns.
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Fixed, preselected training points can miss regions that are difficult to resolve. The SIAM study on failure-informed adaptive sampling proposes estimating failure regions from residuals and enriching collocation points there. The 2023 R3 sampling paper describes a related problem: poor sampling can hinder propagation of solution information from initial or boundary points into the interior.
Which intervention fits the failure signal?
| Observed signal | Intervention to test | Evidence and trade-off |
|---|---|---|
| One loss term stalls or its gradients dominate the others. | Test adaptive loss weighting or gradient-statistics-based learning-rate annealing; compare per-term losses and residual maps before and after. | The 2021 SIAM paper proposes annealing based on gradient statistics for PINN gradient imbalance. It is a research-supported option, not a universal cure: paper. |
| Residual peaks are concentrated in particular regions. | Enrich the collocation set in poorly resolved regions using a residual-informed strategy, while retaining points across the broader domain. | Failure-informed adaptive sampling supports enrichment in residual-defined failure regions; the paper does not establish a universal threshold or point count: paper. |
| Errors suggest the solution is not propagating from constrained points into the interior. | Consider a sampling scheme that retains useful points while resampling or releasing others, and continue checking residuals beyond the current peaks. | The 2023 R3 work proposes retain-resample-release sampling. It cautions that chasing only the highest-residual points can oscillate between peaks and cause the model to forget other regions: paper. |
| Loss balancing and sampling changes do not address persistent architectural optimization problems. | Benchmark a different architecture against the same problem and evaluation protocol before adopting it. | A 2026 ICML paper proposes Physics-Informed Residual Flows to target gradient shattering and flow mismatch; it does not establish a default architecture for Navier–Stokes PINNs: paper. |
| Opposing gradients from PDE and boundary constraints appear to be the issue. | Investigate constraint-alignment methods as a research option, and measure each constraint independently. | A separate 2026 ICML paper discusses aligned constraints and limitations of adaptive weighting and hard constraints in the settings it studies. Its findings do not establish a general fix for Navier–Stokes cases: paper. |
When comparing interventions, hold the formulation, training budget and evaluation protocol constant where possible. Record the failure signal each change targets, whether domain-wide coverage is retained, the added complexity, and the separate validation results. Otherwise, an apparent improvement may reflect a changed evaluation or a trade-off between constraints rather than a better flow solution.
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How should you validate a repaired model?
After changing weights, sampling or architecture, evaluate on held-out points across the domain. Keep the following checks separate:
- PDE residual over space and time, including regions that were not enriched during training.
- Boundary-condition and initial-condition errors, rather than only their contribution to total loss.
- Physical flow quantities relevant to the problem, compared with trusted reference data when available.
Training loss is evidence about the objective being optimized, not by itself proof that the predicted flow is accurate. Compare candidate interventions on the same checks and retain broad-domain evaluation even when adaptive sampling focuses additional effort in high-error regions.
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