ODE filters are a class of probabilistic numerical methods for solving ordinary differential equations (ODEs) that frame the problem of finding a solution to an initial value problem as a Bayesian inference task. Solutions are modeled as probability distributions that also account for the discretization error introduced through the numerical approximation. This probabilistic treatment provides an computation-aware alternative to classical numerical ODE-solvers, which typically only provide point-wise error bounds. It further enables sampling of joint trajectories from the posterior, as well as quantifying uncertainty about the underlying ODE itself. ODE filters offer a flexible framework for incorporating additional information, such as measurements or conservation laws. The general ODE filtering procedure consists of two steps: The first is the definition of a Prior distribution, for the ODE solution, in the form of a stochastic processes, more specifically, as Gauss–Markov processes. Discretization results in nonlinear Gaussian state space models. Second, use a general Bayesian filtering and smoothing algorithm, from the field of recursive Bayesian estimation, to find numerical solutions to the ODE. The naming convention of ODE filters typically reflects the underlying filtering algorithm: For instance, combining the particle filter with this framework yields the particle ODE filter. Since smoothers are extensions of filters, the literature often refers to both under the umbrella term "ODE filters".
Theory
Problem setup Consider a first-order ordinary differential equation (ODE) initial value problem (IVP) of the form
y ˙ ( t ) = f ( y ( t ) , t ) , t ∈ [ 0 , T ] , y ( 0 ) = y 0 ∈ R d . {\displaystyle {\dot {y}}(t)=f(y(t),t),\quad t\in [0,T],\quad y(0)=y_{0}\in \mathbb {R} ^{d}.}
The vector field f : R d × [ 0 , T ] → R d {\textstyle f:\mathbb {R} ^{d}\times [0,T]\rightarrow \mathbb {R} ^{d}} is assumed to be Lipschitz-continuous such that a unique global solution to this ODE-IVP exists according to the Pickard-Lindelöf theorem:. Classical Numerical ODE Solvers, are algorithms that compute approximate solutions y ^ ( t ) ≈ y ( t ) {\textstyle {\hat {y}}(t)\approx y(t)} on a discrete mesh { t n } n = 0 N , 0 = t 0 < ⋯ < t N = T , n ∈ N {\textstyle \{t_{n}\}_{n=0}^{N},\quad 0=t_{0}<\cdots <t_{N}=T,\quad n\in \mathbb {N} } . Probabilistic ODE Solvers additionally estimate the uncertainty introduced through the discretization. ODE Filters are a type of probabilistic ODE solvers, that adopt a Bayesian Inference framework to compute a posterior
p ( y ( t ) ∣ y ( 0 ) , { y ˙ ( t n ) = f ( y ( t n ) , t n ) } n = 0 N ) . {\displaystyle p\left(y(t)\mid y(0),\{{\dot {y}}(t_{n})=f(y(t_{n}),t_{n})\}_{n=0}^{N}\right).}
This is done in two steps. First, prior data and likelihood is defined by modeling solutions to the ODE as a Gauss-Markov process. Second, the posterior is computed with Bayesian filtering and smoothing algorithms.
Step 1: Gauss-Markov process
Prior The prior is specified by a linear time invariant (LTI) stochastic differential equation (SDE) of the form
X ( 0 ) ∼ N ( μ 0 , Σ 0 ) {\displaystyle X(0)\sim {\mathcal {N}}(\mu _{0},\Sigma _{0})}
d X ( t ) = F X ( t ) d t + L d B t . {\displaystyle \mathrm {d} X(t)=FX(t)\mathrm {d} t+L\mathrm {d} B_{t}.}
It describes a stochastic process ("the system"):
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