This framework enables the reliable and highly efficient deployment of scientific surrogates without requiring ground-truth data and establishes computable sufficient conditions that rigorously certify exact decisions for strongly monotone discretizations.
Abstract
Selecting the optimal neural-operator prediction during deployment is challenging when high-fidelity reference solutions are unavailable. We demonstrate that under a squared Hilbert-space loss, ranking a finite model library depends strictly on the low-dimensional span of candidate differences, allowing us to score all models simultaneously using a single anchor-based linearized response of the governing equation. This shared physical diagnostic accurately recovered over 99.6\% of pairwise preferences and 99.0\% of optimal checkpoints across diverse Fourier and convolutional operator libraries for fluid, reaction-diffusion, and wave dynamics. Furthermore, the corrected physical proxy frequently outperformed the best individual candidates, and we establish computable sufficient conditions that rigorously certify exact decisions for strongly monotone discretizations. By exploiting the local dynamical response rather than raw defect magnitude, this framework enables the reliable and highly efficient deployment of scientific surrogates without requiring ground-truth data.
Physics-informed neural networks (PINNs) are commonly evaluated via a single aggregate residual, assuming a smaller residual indicates a better solution. Testing this directly across three constrained PDE systems, I find this assumption can systematically fail. In matched pairs of solvers differing only in whether a defining structural identity is hard-wired or penalized, the penalized variant frequently attains a lower equation residual while violating that identity by several orders of magnitude, causing the exact variant to be falsely ranked worse. Over 64 matched pairs spanning two systems, four network variants, and eight seeds, this inversion occurs in 83\% of cases (95\% Wilson CI: 72--90\%), with rates from 72\% to 94\% across systems. Testing across six architectures--MLP, cPINN, XPINN, hp-VPINN, and physics-informed DeepONet and FNO--inverts the ranking in 46 of 48 pairs, indicating that this variability is problem-dependent rather than specific to the approximator. A third, larger vorticity--streamfunction problem shows the same ordering: the residual-optimal solver violates its structural identity by over six orders above tolerance, despite a residual margin of only 9.37%. Because a scalar loss cannot expose this, I introduce a lexicographic admissibility gate spanning the structural identity, boundary trace, and a solvability integral that must vanish independently of the equation residual. All three must pass before residuals can compete. This gate catches three artifact classes but misses a fourth: a prescribed-structure prior yields fields that pass every single-run check, yet deleting the source term reveals that 97\% of the reported structure survives removal of the physics. Reference data and figures accompany the paper.
This work benchmarks eleven transfer-learning surrogates against four GP methods under an identical selection rule, fidelity budget, and model size, across nine tasks spanning synthetic functions to real chemistry and materials problems, where transfer-learning surrogates reach substantially better solutions using far less computation.
Jaewook Lee, Ethan Errington, Christian D. Lorenz et al.· 0 citations
Neural PDE operators are increasingly trained on reusable solver archives, yet validation often relies on clean prediction error and parameter-agnostic plausibility checks. We introduce cross-parameter relinking, a data-poisoning primitive that makes a triggered input select a valid solution from the same PDE family under an incorrect physical parameter. We term this a wrong-physics backdoor: the output remains physically plausible but is wrong for the intended parameter. The attack exploits tensor-to-parameter provenance failures in multi-parameter archives by stamping the surrogate input and relinking its supervision to a cached alternate-parameter solution for the same latent sample. Across 476 attack campaigns, we evaluate Burgers, advection-diffusion, two-dimensional Navier-Stokes, and an elliptic Poisson case. Fourier Neural Operators and DeepONet provide the primary evidence, with Transformer, GRU, and LSTM models as support. FNO reaches a backdoor success rate of 1.0000 on both advection-diffusion and two-dimensional Navier-Stokes while retaining low clean relative L2 error. Clean-label, label-only, and shuffled controls show that high attack success alone is insufficient: successful attacks must move predictions toward the intended alternate-physics target while preserving bounded clean error. These results expose a structural validation gap: smoothness or generic solver-like behavior is insufficient unless the provenance of the intended physical parameter is also verified.
Reliable forecasting of nonlinear physical systems underpins scientific discovery and engineering decision-making. Yet high-fidelity simulations are prohibitively costly, and machine-learning surrogates can be opaque and encode assumptions about system dynamics, limiting generalizability. Pretrained transformers mapping synthetic ODE trajectories to equations offer interpretable alternatives, promising transfer without system-specific equation knowledge. Transferring them reliably to high-dimensional physical data, however, remains an open challenge. We develop a verifier-guided (VG) workflow around ODEFormer as a symbolic backbone, using dynamical and physical-admissibility criteria to select from a multi-trajectory candidate equation pool, enabling transfer. On canonical Van der Pol oscillators, VG outperforms the original ODEFormer workflow across held-out initial conditions. We then address vortex shedding, a phenomenon occurring in atmospheric and plasma systems of societal relevance, through coordinate reduction and symbolic discovery at fixed and varying Reynolds numbers. VG discovers fixed-parameter reduced-order equations that recover the fundamental shedding oscillator and higher harmonics without a wake-specific candidate library or prescribed Navier-Stokes structure, while the cross-parameter model generalizes to withheld regimes. Reconstruction fidelity alone did not determine symbolic discoverability, highlighting the importance of compatibility between latent dynamics and the backbone's pretraining distribution. This work establishes a verifier-guided neural-to-symbolic methodology for interpretable and physically auditable forecasting in the natural sciences.
Physics-informed neural networks (PINNs) approximate partial differential equations (PDEs) by enforcing governing equations and boundary conditions during training, but their accuracy depends on how collocation points are distributed and updated. We propose spatiotemporal compositional active sampling (STCAS), a reference-assisted offline configuration procedure that uses an analytic or high-accuracy numerical solution to rank complete three-stage sampling plans. It screens eight fixed rules, forms a task-specific shortlist, and evaluates bounded fixed, switched, and locally blended plans with independent selection sets and a composition guard. A safety-anchor decision retains the standard PINN unless the selected candidate is at least 5% better. Across five evaluations on 18 analytically specified two-dimensional Poisson tasks, this protocol improves 16 task means and ties two, reducing aggregate relative-L2 error by 12.8% (hierarchical-bootstrap 95% interval [6.78%,19.32%]; one-sided paired Wilcoxon p=2.19×10−4). Against the confirmed fixed plan, aggregate error decreases by 8.2%. In comparison experiments designed for two transfer tasks and matched for main PINN training budgets, STCAS achieves the lowest aggregate mean reported error among the compared methods for both a steady convection–diffusion equation and a nonlinear time-dependent Burgers equation; its offline search cost is additional.
Ju-Zheng Zhang, Shi-Yang Li, Tao Zhu et al.· Mathematics· 0 citations
Recent benchmarks show that deep-learning models for perturbation prediction do not outperform simple baselines operating in principal-component (PCA) space. We explain this with an information-theoretic ceiling: for any orthonormal projection basis Φ, the squared correlation between prediction and truth is bounded by the variance the basis explains (r2 ≤ VE), so no model complexity can recover signal discarded at projection. On chemical perturbations (sciPlex3, LINCS L1000), the eigenbasis of a gene association network captures only 10–12% of drug-response variance and yields chance-level predictions, while PCA captures 90–99%. Graph wavelets built on the same network recover ∼88%, localising the drug signal in high-frequency modes that the standard low-pass eigenbasis discards. On CRISPRa genetic perturbations the ranking inverts: the network basis outperforms PCA across all dimensions tested. Controls on topology, null networks and data leakage confirm the effect is structural. The right basis depends on the perturbation modality: PCA captures the variance that drives chemical responses, the network basis captures the cascade structure that drives genetic ones, and bases that access the network’s full graph spectrum (such as graph wavelets) recover both from the same topology.