CONFIDE: Contextual Finite Differences Modelling of PDEs

TL;DR

This work tackles inferring context-dependent PDE coefficients when only partial mechanistic knowledge is available. It introduces CONFIDE, a context-aware hybrid that encodes dynamics with an autoencoder and jointly estimates coefficient functions which are then used by a standard PDE solver to forecast future states, achieving zero-shot generalization across unseen coefficient contexts. The method combines an initial-conditions aware autoencoder with a PDE-consistency loss, enabling accurate coefficient reconstruction and robust predictions across multiple PDE families (constant-coefficient, Burgers', FitzHugh-Nagumo, Navier–Stokes) and under out-of-distribution scenarios. Experiments show that CONFIDE consistently outperforms Neural ODE, FNO, Unet, and DINo baselines, while offering explainability through the recovered coefficient functions and providing significant speed advantages over integration-based approaches. The approach holds promise for data-efficient, physics-informed modeling in domains like battery dynamics, where different instances exhibit varying coefficients within a shared PDE structure.

Abstract

We introduce a method for inferring an explicit PDE from a data sample generated by previously unseen dynamics, based on a learned context. The training phase integrates knowledge of the form of the equation with a differential scheme, while the inference phase yields a PDE that fits the data sample and enables both signal prediction and data explanation. We include results of extensive experimentation, comparing our method to SOTA approaches, together with ablation studies that examine different flavors of our solution.

Figures (9)

• Figure 1: (a) Inference process: given an observed spatio-temporal signal, CONFIDE estimates the PDE coefficients that best describe it. These can be plugged into a PDE solver together with the known operator form $F$, and an initial condition (dashed line) to obtain a prediction of the signal for future time-steps. (b) Training process: In each iteration, CONFIDE observes a set of signals generated by the same family of PDEs. For each train signal, CONFIDE evaluates the PDE coefficients best describing the observed signal, and all the spatio-temporal derivatives that are known to be in the functional form of the PDE (e.g., $\frac{\partial u}{\partial t}$, $\frac{\partial^2{u}}{\partial{x}^2}$, ...). The derivatives and coefficients are then plugged into the operator $F$ which is then used to minimize the functional loss (as in Eq. \ref{['eq:coefficient_estimator_loss']}) and train a context-based coefficient estimator.
• Figure 2: Constant coefficients (Section \ref{['ssec:pde_exp1']}). (a) Prediction error vs. prediction horizon, for different algorithms. CONFIDE, in red, is our approach. (b) Estimated value of the $\partial^2 u / \partial x^2$ coefficient vs. ground truth, for test set ($R^2 = 0.93$).
• Figure 3: Burgers' PDE: (a) A solution of the Burgers’ equation. The black plot in each figure displays the ground truth. Rows correspond with the predicted solution by the respective algorithm (top row for CONFIDE displayed in red). Each column shows the solution at a different time point. The rightmost column shows the solution at t = 100 zoomed to demonstrate the differences. (b) Estimation of the coefficient function $b(x,t,u)$ of the Burgers' equation from \ref{['eq:pde_exp2']}. CONFIDE manages to accurately estimate the spatio-temporal dynamics of the coefficient, based on a context ratio of $\rho=0.2$.
• Figure 4: 2D-FitzHugh-Nagumo PDE: Figures in the top row show the ground truth of $R_v$ for different time points, and the rows below show the estimation of it by the different approaches. CONFIDE estimates $R_v$ directly and near-perfectly recovers the unknown part of the PDE even as the prediction horizon increases. For the other algorithms we evaluated $R_v=u-v$ from the predictions of $u$ and $v$.
• Figure 5: 2D-FitzHugh-Nagumo PDE: prediction error as horizon increases, for different approaches.