A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting

1. Problem Statement

The paper addresses the nonlinear Bayesian filtering problem in high-dimensional continuous-state, discrete-observation settings.

• The Challenge: Traditional filtering methods (e.g., Kalman filters) rely on unimodality assumptions, while Sequential Monte Carlo methods (Particle Filters) suffer from the curse of dimensionality, where the number of particles required to maintain accuracy grows exponentially with the state dimension $d$.
• The Goal: To approximate the filtering probability density function $p_k(x | y_{0:k})$ (the distribution of the hidden state $S_t$ given noisy observations $y_{0:k}$) in high dimensions ($d \ge 10$) with rigorous convergence guarantees.
• Mathematical Formulation:
  • State Dynamics: $dS_t = \mu(S_t)dt + \sigma(S_t)dB_t$ (Stochastic Differential Equation).
  • Observations: Discrete measurements $O_k \sim g(O_k | S_{t_k})$.
  • Evolution: The unnormalized filtering density evolves via a Fokker–Planck equation (prediction step) and is updated via Bayes' formula (update step).

2. Methodology: Deep Splitting with Feynman–Kac

The authors propose a Deep Splitting scheme that combines probabilistic numerical methods with deep learning. The algorithm operates in a prediction-update cycle:

A. Prediction Step (Fokker–Planck Approximation)

Instead of solving the high-dimensional Partial Differential Equation (PDE) directly on a grid, the method uses a Feynman–Kac representation.

• Time Discretization: The interval between observations is split into $N$ sub-steps.
• Operator Splitting: The adjoint generator $A^*$ of the diffusion process is decomposed into a second-order operator $A_b$ (associated with an auxiliary process $X$ having drift $b$) and a first-order operator $F_b$.
• Recursive Approximation: The solution is approximated by a sequence of functions $\pi_{k,n}$ defined recursively:
  $$\pi_{k,n+1}(x) = \mathbb{E} \left[ G_b \pi_{k,n}(X_{t_{N-n+1}}^{x}) \right]$$
  where $G_b$ is a first-order differential operator approximating the integral term, and $X$ is an auxiliary diffusion process.

B. Optimization Formulation (Deep Learning)

The conditional expectation in the Feynman–Kac formula is reformulated as a minimization problem solvable by neural networks:

• Objective: Find a function $u$ (parameterized by a neural network) that minimizes the $L^2$ error between the network output and the target term derived from the previous step.
  $$\hat{\pi}_{k,n+1} = \arg \min_{u} \mathbb{E} \left[ |u(Z_{N-(n+1)}, Y) - G_b \hat{\pi}_{k,n}(Z_{N-n}, Y)|^2 \right]$$
• Architecture: Uses Fully Connected Neural Networks (FCNNs) with Energy-Based activation functions (exponential of a scalar output). This ensures the output is positive (required for probability densities) and avoids the need to compute the normalizing constant during training.
• Online Capability: Unlike offline methods that retrain for every observation path, this scheme learns a function over the space of all possible observation sequences, enabling online inference for new data post-training.

C. Update Step

At discrete observation times $t_k$, the predicted density is updated using Bayes' formula:
$$\pi_{k,0}(x) = \pi_{k-1,N}(x) \cdot L(y_k, x)$$
where $L$ is the measurement likelihood.

3. Key Contributions

1. Rigorous Convergence Analysis: The paper establishes a strong convergence rate of order 1 ($O(\tau)$) in the $L^\infty$ norm with respect to the time step $\tau$, under a parabolic Hörmander condition. This is a significant theoretical advancement, as many deep learning PDE solvers lack such rigorous error bounds.
2. Decoupling from Filtering: As a corollary, the authors prove the convergence rate for the approximation of the Fokker–Planck equation alone, independent of the filtering update steps.
3. High-Dimensional Scalability: The method is designed to mitigate the curse of dimensionality by using Monte Carlo sampling and neural networks, avoiding grid-based discretization.
4. Auxiliary Drift Selection: The authors introduce a specific choice for the auxiliary drift function $b$ (Eq. 5 in the paper) that eliminates first-order terms in the error analysis, facilitating the proof of the $O(\tau)$ convergence rate.

4. Numerical Results

The authors validate the method through numerical experiments:

• 1D Convergence Study:
  • Tanh Process: A system satisfying the paper's regularity assumptions. The method demonstrated a convergence order close to 1, matching the theoretical prediction.
  • Bistable Process: A system with weaker regularity assumptions. The convergence order dropped to approximately 0.5, highlighting the importance of the theoretical assumptions for optimal performance.
• 10-Dimensional Nonlinear Example:
  • A high-dimensional system with coupled tanh drifts.
  • Comparison: The proposed method (EBDS) was compared against an Ensemble Kalman Filter (EnKF) and a Bootstrap Particle Filter (PF).
  • Performance: The deep splitting method achieved accuracy comparable to the EnKF (which assumes near-Gaussianity) and significantly outperformed the Particle Filter with $10^5$ particles. To match the deep splitting method's performance, the Particle Filter required $\approx 10^6$ particles, demonstrating the efficiency of the proposed approach in high dimensions.
  • Metrics: Evaluated using Mean Absolute Error (MAE) and Negative Log-Likelihood (NLL). The method captured the shape of the density (NLL) more accurately than the mean (MAE).

5. Significance

• Theoretical Rigor: This work bridges the gap between deep learning-based PDE solvers and classical numerical analysis by providing a provable convergence rate for a filtering problem, a rare achievement in the field of AI for science.
• Practical Applicability: The method offers a viable solution for high-dimensional filtering problems (e.g., weather forecasting, finance, robotics) where traditional particle filters fail due to computational cost and grid-based methods are infeasible.
• Online Inference: By learning a universal function over observation sequences, the method enables real-time filtering without retraining, making it suitable for dynamic environments.
• Generalization: The framework is flexible enough to handle various drift and diffusion coefficients and can be adapted to different observation models, provided the underlying regularity conditions are met.

In summary, the paper presents a mathematically grounded, deep learning-enhanced numerical scheme that successfully solves the nonlinear filtering problem in high dimensions with established convergence guarantees, offering a robust alternative to existing particle and grid-based methods.