Discovering Latent Response Laws in Forced Physical Systems

Numerical site-response predictions often deviate from observations, yet correcting these discrepancies is difficult because records are limited in both sensor coverage and number of events. This study proposes the Transfer-Enabled Forced Latent Autoencoder for Response Equations (FLARE-T) to improve these predictions by learning and calibrating low-dimensional latent dynamics that connect the base acceleration input to acceleration outputs at multiple depths. FLARE-T learns a low-dimensional response manifold and input-driven dynamics from dense finite-element simulations. It then trains a sparse encoder to map simulated sensor responses into the learned coordinates and uses limited records to calibrate the dynamics within them. A short response window initializes each prediction, while the complete base motion drives the response. The framework was evaluated using a layered-soil centrifuge test and the Lotung field vertical array. Test-set results show that FLARE-T improved multi-depth acceleration histories and 5%-damped pseudoacceleration response spectra relative to the original finite-element models, reducing errors at every evaluated sensor for motions of different intensities and, at Lotung, for both horizontal components. Two Lotung source models with different constitutive parameters achieved comparable test-set accuracy, indicating reduced dependence on precise prior calibration. FLARE-T therefore provides a data-efficient means of combining dense numerical response information with limited field records to improve future site-response predictions.

Identifying latent dynamical systems from noisy, high-dimensional measurements is a central problem at the intersection of representation learning, system identification, and scientific discovery. We present DYSCO, a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and the governing dynamics from such observations, by leveraging multiple independent noisy views of the same underlying process to disentangle signal from noise. By parameterizing the dynamics in a structured functional basis, our framework further enables symbolic recovery of the governing equations within an affine gauge. We offer theoretical guarantees for strong identification up to an affine indeterminacy, extending prior identifiability results to the realistic setting of noisy nonlinear observations. Empirically, we demonstrate accurate recovery of both latent trajectories and flow fields across a diverse set of dynamical regimes (e.g., chaotic, oscillatory, and metastable) under both Gaussian and Poisson observation noise, the latter being particularly relevant for neural recordings.

Learning Koopman operators with autoencoders enables linear prediction in a latent space, but long-horizon rollouts often drift off the learned manifold, leading to phase and amplitude errors on systems with switching, continuous spectra, or strong transients. We introduce two complementary components that make Koopman predictors more robust. First, we add an attention-free latent memory (AFT) block that aggregates a short window of past latents to produce a corrected latent before each Koopman update. Unlike multi-head attention, AFT operates in linear time and adds only $\approx$30k parameters ($3d^2 + T^2$, fewer than matched multi-head attention), yet captures the local temporal context needed to suppress error divergence. Second, we propose dynamic re-encoding: lightweight, online change-point triggers (EWMA, CUSUM, and sequential two-sample tests) that detect latent drift and project predictions back onto the autoencoder manifold. Across three benchmark systems -- Duffing oscillator, Repressilator, IRMA -- our model consistently reduces error accumulation compared to a Koopman autoencoder and matched-capacity multi-head attention. We also compare against GRU and Transformer autoencoders, evaluated both from initial conditions and with a 50-step context, and find that Koopman+AFT (with optional re-encoding) attains markedly lower long-horizon error while maintaining lower inference latency. We report improvements over horizons up to 1000 steps, together with ablations over trigger policies. The result is a fast, compact predictor that stays on the learned manifold over long horizons.