Open Problems

Scientific ML: PDEs & Neural Operators

Systematic Generalisation and Robustness Evaluation of Neural Operators in 3D PDE Systems

Scope to testOpen
Possible candidate · 3/5 runs6 papers report this50% from 2025+

Generated automatically from the limitations stated in 6 papers (NeurIPS, ICLR, ICML), listed under Evidence. It is not a paper, and it does not come from papers submitted to CSPaper.

The problem

Current neural operator architectures are evaluated almost exclusively on 2D planar domains, leaving their predictive accuracy and robustness in 3D systems completely unmeasured. Without multi-dimensional benchmarks and controlled cross-dimensional studies, it is unknown whether empirical performance, discretization invariance, and error characteristics observed in 2D transfer to three-dimensional physical systems. Consequently, researchers and practitioners cannot determine whether 2D results provide reliable evidence for full 3D simulation tasks.

Why it matters

Provides the first empirical ground truth on how neural operator models generalize from 2D to 3D physical systems, establishing reliable baselines for high-dimensional scientific machine learning.

Ways to approach it

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  1. 1

    Curate a standardized suite of paired 2D and 3D PDE benchmarks (such as Navier-Stokes, Poisson, and wave propagation problems) with comparable boundary condition variability, measuring relative $L_2$ error, training stability, and sample efficiency across leading neural operator models (e.g., FNO, DeepONet, and message-passing neural operators).

  2. 2

    Measure compute, memory footprint, and scaling trade-offs as spatial discretization extends from 2D slices to 3D volumes, evaluating whether standard architectural parameterizations maintain their representational capacity under constrained compute budgets.

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Why it might fail

Generating and training high-resolution 3D PDE datasets may exceed available academic compute budgets, or concurrent community benchmark releases may standardize 3D evaluations before the study is complete.

Evidence

Each paper's own statement of the limitation, verbatim.

Nearest existing work

Related open problems

Scientific ML: PDEs & Neural Operators

Barrier to removePartly addressed

Semi-Supervised Neural Operator Learning for Parameter-Scarce Trajectories

Existing neural operators and trajectory surrogate models for continuum mechanics require dense, ground-truth physical parameter labels for every training trajectory. In experimental and real-world physical systems, recording state trajectories is feasible, but measuring underlying continuum or constitutive parameters is often costly, destructive, or impossible. Because existing methods lack semi-supervised formulations, they cannot leverage unlabeled experimental trajectories and remain restricted to fully labeled synthetic simulations.

Possible candidate · 2/5 runs4 papers report this75% from 2025+
Generated automatically, not curated by hand. Automated prior-work checks catch about a third of existing work, so treat this problem as a lead to investigate.