# Principal Component Analysis

In ESM2 residue embeddings, polar/nonpolar binary patterns organize into a readable geometry. Generating 50 sequences for each of all 128 heptad binary patterns, 6400 in total, and running position-preserving PCA yields a jellyfish-shaped manifold: hydrophobic content spreads along PC1, while PC2 and PC3 encode heptad phase as near sine and cosine. A charge alphabet stays ordered, but scrambled amino-acid groupings or the non-protein model ByT5 lose the cyclic organization.

In ESM2 residue embeddings, polar/nonpolar binary patterns organize into a readable geometry. Generating 50 sequences for each of all 128 heptad binary patterns, 6400 in total, and running position-preserving PCA yields a jellyfish-shaped manifold: hydrophobic content spreads along PC1, while PC2 and PC3 encode heptad phase as near sine and cosine. A charge alphabet stays ordered, but scrambled amino-acid groupings or the non-protein model ByT5 lose the cyclic organization.

DINGO-TGR infers binary parameters and six post-Newtonian phase-deformation parameters jointly with neural posterior estimation, then reweights the network's approximate posterior by importance sampling. For GW150914 the inference takes about 38 minutes against roughly 177 hours for bilby-tgr with dynesty, and about 4.4 hours against 238 hours for the higher signal-to-noise GW250114. GW150914 shows a mild offset from general relativity in the leading principal-component mode, but 115 of 1,000 general-relativity injections resembling GW150914 reach or exceed that offset, an empirical upper-tail p-value of about 0.115 that does not reject the general-relativity null at the 5% level.

DINGO-TGR infers binary parameters and six post-Newtonian phase-deformation parameters jointly with neural posterior estimation, then reweights the network's approximate posterior by importance sampling. For GW150914 the inference takes about 38 minutes against roughly 177 hours for bilby-tgr with dynesty, and about 4.4 hours against 238 hours for the higher signal-to-noise GW250114. GW150914 shows a mild offset from general relativity in the leading principal-component mode, but 115 of 1,000 general-relativity injections resembling GW150914 reach or exceed that offset, an empirical upper-tail p-value of about 0.115 that does not reject the general-relativity null at the 5% level.

Quasar variability is usually tested for periodic signals against a damped random walk (DRW) null, and a mis-specified noise model biases the significance estimates. Nullasar uses flow matching to learn the statistical structure of stochastic variability from an ensemble of synthetic light curves without assuming a noise model, then generates null realizations conditioned on a single observed curve. On synthetic AR(2) light curves its periodicity-detection AUC is 0.9648, averaged over three independently trained models, above 0.9180 for fitting AR(2) per curve; in the AR(1) case it matches fitting the correct DRW model.

Quasar variability is usually tested for periodic signals against a damped random walk (DRW) null, and a mis-specified noise model biases the significance estimates. Nullasar uses flow matching to learn the statistical structure of stochastic variability from an ensemble of synthetic light curves without assuming a noise model, then generates null realizations conditioned on a single observed curve. On synthetic AR(2) light curves its periodicity-detection AUC is 0.9648, averaged over three independently trained models, above 0.9180 for fitting AR(2) per curve; in the AR(1) case it matches fitting the correct DRW model.

Geological uncertainty in subsurface flow systems is often hierarchical: a set of geological metaparameters controls overall statistics while cell-by-cell property fields vary, and both must be estimated from observations. An adaptive-tempered sequential Monte Carlo method estimates metaparameters and Gaussian latent variables jointly within one likelihood framework, with surrogate models supplying all forward evaluations. Across four synthetic true models, its posterior metaparameter distributions and flow predictions agree closely with reference MCMC results, while single-GPU data assimilation is 2.5–7.0 times faster and needs fewer function evaluations.

Geological uncertainty in subsurface flow systems is often hierarchical: a set of geological metaparameters controls overall statistics while cell-by-cell property fields vary, and both must be estimated from observations. An adaptive-tempered sequential Monte Carlo method estimates metaparameters and Gaussian latent variables jointly within one likelihood framework, with surrogate models supplying all forward evaluations. Across four synthetic true models, its posterior metaparameter distributions and flow predictions agree closely with reference MCMC results, while single-GPU data assimilation is 2.5–7.0 times faster and needs fewer function evaluations.

Nuclear physics has long relied on phenomenological models to predict nuclear binding energies, while high-precision AI models are hard to trust when extrapolating. An interpretable neural network taking only proton and neutron numbers as input spontaneously forms a double-helix geometry in its internal representations, reproduces macroscopic terms such as the liquid-drop model in a principal-component decomposition, and rediscovers the local Jaffe factorization proposed in 1969 and since overlooked. Applied as a local correction to the WS4 model, this factorization brings the root-mean-square binding-energy error below 0.12 MeV without relying on a black box.

Nuclear physics has long relied on phenomenological models to predict nuclear binding energies, while high-precision AI models are hard to trust when extrapolating. An interpretable neural network taking only proton and neutron numbers as input spontaneously forms a double-helix geometry in its internal representations, reproduces macroscopic terms such as the liquid-drop model in a principal-component decomposition, and rediscovers the local Jaffe factorization proposed in 1969 and since overlooked. Applied as a local correction to the WS4 model, this factorization brings the root-mean-square binding-energy error below 0.12 MeV without relying on a black box.

Equivalent-circuit fitting of electrochemical impedance spectra has depended on a human choosing a circuit topology and starting values, even though different topologies can produce nearly identical spectra. A University of Zagreb team expanded a few manually fitted spectra into a synthetic reference library, projected it into principal-component space, and used nearest-neighbour retrieval to shortlist topologies and supply physically plausible starting values at once, then refined each candidate by complex nonlinear least squares and chose by fit quality. Against independent manual fits, most parameters agreed within 1%, reduced chi-squared was lower in every case, and one spectrum took about 5 to 6 seconds.

Equivalent-circuit fitting of electrochemical impedance spectra has depended on a human choosing a circuit topology and starting values, even though different topologies can produce nearly identical spectra. A University of Zagreb team expanded a few manually fitted spectra into a synthetic reference library, projected it into principal-component space, and used nearest-neighbour retrieval to shortlist topologies and supply physically plausible starting values at once, then refined each candidate by complex nonlinear least squares and chose by fit quality. Against independent manual fits, most parameters agreed within 1%, reduced chi-squared was lower in every case, and one spectrum took about 5 to 6 seconds.