Deep LPPLS: Forecasting of temporal critical points in natural, engineering and financial systems

TL;DR

The LPPLS framework models finite-time regime changes via nonlinear parameters $t_c$, $m$, and $\omega$ in the observable $\mathcal{O}(t)=A+B(t_c-t)^m+C(t_c-t)^m\cos(\omega\ln(t_c-t)-\phi)$, but conventional calibration like Levenberg–Marquardt can be slow and unstable. The authors introduce two neural calibration strategies: Mono-LPPLS-NN (M-LNN) for bespoke per-series fitting, and Poly-LPPLS-NN (P-LNN) trained on large synthetic LPPLS sequences with varied noise (white and AR(1)) to enable fast, out-of-sample estimation on fixed-length inputs. Across synthetic data, both NN approaches outperform LM in estimating the nonlinear parameters $t_c$, $m$, and $\omega$, with first-order stochastic dominance in their error distributions; P-LNN offers substantial speedups, while M-LNN provides strong per-series fidelity. Empirical tests on three real-world datasets (Dot-com bubble, silver bubble, and a rockslide) demonstrate improved consistency and accuracy in predicted critical times, highlighting the practical potential for near real-time forecasting of regime changes in finance and geophysics. This work bridges deep learning with the prediction of finite-time singularities, enabling faster, robust inference and opening avenues for richer noise modeling and more flexible architectures in future research.

Abstract

The Log-Periodic Power Law Singularity (LPPLS) model offers a general framework for capturing dynamics and predicting transition points in diverse natural and social systems. In this work, we present two calibration techniques for the LPPLS model using deep learning. First, we introduce the Mono-LPPLS-NN (M-LNN) model; for any given empirical time series, a unique M-LNN model is trained and shown to outperform state-of-the-art techniques in estimating the nonlinear parameters $(t_c, m, ω)$ of the LPPLS model as evidenced by the comprehensive distribution of parameter errors. Second, we extend the M-LNN model to a more general model architecture, the Poly-LPPLS-NN (P-LNN), which is able to quickly estimate the nonlinear parameters of the LPPLS model for any given time-series of a fixed length, including previously unseen time-series during training. The Poly class of models train on many synthetic LPPLS time-series augmented with various noise structures in a supervised manner. Given enough training examples, the P-LNN models also outperform state-of-the-art techniques for estimating the parameters of the LPPLS model as evidenced by the comprehensive distribution of parameter errors. Additionally, this class of models is shown to substantially reduce the time to obtain parameter estimates. Finally, we present applications to the diagnostic and prediction of two financial bubble peaks (followed by their crash) and of a famous rockslide. These contributions provide a bridge between deep learning and the study of the prediction of transition times in complex time series.

Figures (10)

• Figure 1: Diagram of the M-LNN architecture and training process. The network takes an input vector of length $n$, representing the time-series data for which the LPPLS parameters are to be estimated. The output layer, comprising three nodes, produces the estimated LPPLS parameters $t_c$, $m$, and $\omega$. These estimates are then utilized to construct the corresponding LPPLS time-series. The Mean Squared Error (MSE) between the input time series ($X$) and the LPPLS time-series is computed as the network's loss function ($\mathcal{L}$), with an additional penalty term to ensure the parameters remain within the designated bounds.
• Figure 2: A diagram of the P-LNN architecture. The network accepts an input vector of length $252$ representing a fixed-length time-series for which you want to obtain the LPPLS estimated parameters. Its output layer consists of three nodes which represent the LPPLS parameters $t_c$, $m$ and $\omega$. Next, we calculate the loss ($\mathcal{L}$) by taking the MSE between the network output and the true LPPLS parameters used to generate the batch of synthetic LPPLS series.
• Figure 3: Cumulative distribution function (CDF) of absolute parameter error per estimation technique organized along rows for three classes of noise used in model training and ordered in four columns according to the variable whose error is quantified.
• Figure 4: Dot-com bubble: the Nasdaq composite index price is plotted as a function of time (dark line) together with the LPPLS fits of the three competing models. The time interval of calibration goes from $t_1$ (green vertical dotted-dashed line) to $t_2$ (vertical red dotted-dashed line) and is represented in grey. We place ourselves in the "present" time $t_2$ so that all data to the right of $t_2$ is not seen and corresponds to the out-of-sample or future evolution that we aim to predict, and in particular, the actual $t_c$. Thus, all PDFs are constructed at time $t_2$. This realised critical time is interpreted to lie between the time of the price peak (black dotted-dashed vertical line) and the trough of its drawdown (black dashed vertical line). This range is indicated in shaded red colour. The M-LNN model (orange line) forecasts reasonably well the critical time, as evidenced from its PDF of $t_c$'s that overlaps largely the interval of the realised drawdown. The LM model (blue line) predicts $t_c$'s that are too late and with more variability in its predictions, reflected in its wider PDF spread. The P-LNN-100K model (purple line) captures the price trend well, with a narrower PDF indicating a higher concentration of $t_c$ predictions that are slightly early compared with the range of the actual critical time.
• Figure 5: 2011 Silver bubble: the silver AGQ price is plotted as a function of time (dark line) together with the LPPLS fits of the three competing models. Same meaning of the different lines as in figure \ref{['fig:dotcom']}. In particular, the "present" time $t_2$ is indicated by the vertical red dotted-dashed line and the goal is to predict the subsequent behavior and especially the realized critical time $t_c$ of the crash. This time is well-defined by the vertical black dotted-dashed line indicating the time when the silver price peaked followed by an abrupt crash. The comparison of the three PDFs shows the significant superior performance of the NN models with the M-LNN model (orange line) providing an excellent prediction.