Differentiable Kalman Filter: Methods & Trends

• Differentiable Kalman Filter is a family of hybrid estimators that embed the classical Kalman recursion within a differentiable framework for end-to-end optimization.
• It leverages efficient forward–backward matrix gradient computations that significantly speed up training, achieving empirical improvements like 53× speedups over classical sensitivity methods.
• DKF methods integrate model-based filtering with learned components to adapt noise models and dynamics in applications like visual-inertial odometry and multi-object tracking.

The differentiable Kalman filter (DKF) is, in the modern machine-learning sense, a Kalman-filter-based estimator whose prediction, update, and uncertainty-propagation operations are embedded in a differentiable computational graph, so that gradients can be propagated through the filter to optimize noise models, measurement models, dynamics operators, or upstream neural-network parameters end to end. In practice, this notion covers a family of hybrid model-based and learned estimators rather than a single canonical algorithm. The term is historically overloaded: “DKF” has also denoted desensitized, discriminative, deep, distributed, and even discrete Kalman filters in different literatures, so precise usage depends on context (Parellier et al., 2023).

1. Terminological scope and acronym ambiguity

The acronym “DKF” does not identify a unique method across the filtering literature. A technically accurate reading therefore begins by disambiguating the major usages.

DKF usage	Core meaning	Representative source
Differentiable Kalman Filter	Kalman recursion treated as a differentiable layer/module for end-to-end optimization	(Wagstaff et al., 2022, Chiu et al., 2023, He et al., 9 Aug 2025, Wu et al., 9 Sep 2025)
Discriminative Kalman Filter	Closed-form filtering based on a learned Gaussian approximation to $p(\text{state}\mid \text{observation})$	(Burkhart et al., 2016, Burkhart, 2018)
Desensitized Kalman Filter	Filter that penalizes sensitivity to uncertain parameters	(Lou, 2015)
Deep Kalman Filter	Neural architecture outputting Gaussian probability measures from sequential data	(Horvath et al., 2023)
Distributed/Discrete Kalman Filter	Networked or implementation-oriented variants unrelated to end-to-end differentiability	(Kettner et al., 2017, Yuan et al., 2017, Ryu et al., 2019, Ryu et al., 2022, Mao et al., 2023, Iqbal et al., 22 Jan 2025, Yang et al., 11 Apr 2025)
Kalman filters on differentiable manifolds	“Differentiable” refers to manifold geometry rather than autodiff-style training	(He et al., 2021)

In the modern learning literature, a differentiable Kalman filter typically preserves the algebraic structure of Bayesian filtering while allowing gradient-based optimization through the filter. This distinguishes it from distributed DKF papers, where the “D” denotes distributed, and from manifold papers, where “differentiable” denotes smooth state spaces rather than trainable computation graphs (Iqbal et al., 22 Jan 2025, He et al., 2021).

2. Differentiating the Kalman recursion

The most direct formulation of a differentiable Kalman filter treats the standard linear-Gaussian recursion as the forward pass and derives an exact backward pass for a scalar loss over the full trajectory. In the notation of the closed-form backpropagation work, the forward recursion is

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$

$P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$

$z_n = y_n - H_n \hat x_{n|n-1},$

$S_n = H_n P_{n|n-1} H_n^T + R_n,$

$K_n = P_{n|n-1} H_n^T S_n^{-1},$

$\hat x_{n|n} = \hat x_{n|n-1} + K_n z_n,$

$P_{n|n} = (I - K_n H_n) P_{n|n-1}.$

A differentiable formulation then specifies a scalar objective

$$L = \sum_{n=1}^{N} \bigl(l_{n|n-1} + l_{n|n}\bigr),$$

and propagates matrix adjoints backward through the recursion to obtain exact gradients with respect to $P_0$, $\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$0, $\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$1, and even the measurements $\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$2 (Parellier et al., 2023).

For the negative log-likelihood

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$3

the local derivatives include

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$4

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$5

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$6

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$7

Because covariance parameters are symmetric, the gradients are symmetrized, and when a covariance is parameterized as $\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$8, the gradient with respect to the factor is

$\hat x_{n|n-1} = A_n \hat x_{n-1|n-1} + B_n u_n,$9

This backward matrix-gradient formulation is computationally important: the paper reports $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$0 complexity for a full matrix derivative, versus $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$1 for classical forward sensitivity equations, with empirical speedups of about $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$2 over sensitivity equations and about $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$3 over PyTorch automatic differentiation in a synthetic $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$4D example (Parellier et al., 2023).

This suggests that the core mathematical content of a modern DKF is not a new filtering equation but a trainable forward–backward pair: a Kalman recursion in the forward pass and an exact or efficient adjoint in the backward pass.

3. Learned models inside the filter

A recurrent design pattern in differentiable Kalman filtering is to retain a structured process model while learning the components that are difficult to specify analytically. In visual-inertial odometry, the self-supervised DKF uses a robocentric EKF with an IMU-driven process model and a neural relative-pose measurement model. The network predicts both a relative pose and a measurement covariance

$P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$5

with diagonal entries parameterized by

$P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$6

The posterior state from the filter, rather than the raw network output, is then used in a self-supervised photometric reconstruction loss. On a visually degraded version of EuRoC, this estimator operated without a significant reduction in accuracy in cases where classical estimators consistently diverged, and it recovered metric scene scale from IMU information (Wagstaff et al., 2022).

In cooperative 3D multi-object tracking, a differentiable multi-sensor Kalman filter is used to learn per-detection observation uncertainty for each connected autonomous vehicle. The tracking state is $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$7-dimensional,

$P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$8

the observation is $P_{n|n-1} = A_n P_{n-1|n-1} A_n^T + Q_n,$9-dimensional,

$z_n = y_n - H_n \hat x_{n|n-1},$0

and the covariance network predicts residual standard deviations whose squares define the diagonal of the measurement covariance. On V2V4Real, the resulting method improved tracking accuracy by $z_n = y_n - H_n \hat x_{n|n-1},$1 with only $z_n = y_n - H_n \hat x_{n|n-1},$2 communication costs compared with the state-of-the-art method (Chiu et al., 2023).

A more adaptive line of work addresses online noise-statistics drift. OTAKNet constructs a source distribution from the filter’s predictive measurement model,

$z_n = y_n - H_n \hat x_{n|n-1},$3

constructs a target distribution from the current observation and a window of past innovations, and minimizes an entropically regularized optimal-transport loss between them. The online update is

$z_n = y_n - H_n \hat x_{n|n-1},$4

with no ground-truth state labels. On the NCLT dataset under limited training, OTAKNet achieved $z_n = y_n - H_n \hat x_{n|n-1},$5 dB MSE versus $z_n = y_n - H_n \hat x_{n|n-1},$6 for KalmanNet (He et al., 9 Aug 2025).

A more ambitious formulation makes the dynamics operator itself trainable. DKFNet treats filtering as an adjoint-based two-level optimization problem: first, a field inversion step optimizes a sequence of transition operators $z_n = y_n - H_n \hat x_{n|n-1},$7 to reduce the residual

$z_n = y_n - H_n \hat x_{n|n-1},$8

and then a neural closure model is trained via

$z_n = y_n - H_n \hat x_{n|n-1},$9

In the reported rocket and Allen–Cahn examples, the method reduced state reconstruction error by at least $S_n = H_n P_{n|n-1} H_n^T + R_n,$0 compared to the classical Kalman filter while maintaining uncertainty quantification (Wu et al., 9 Sep 2025).

4. Related DKF formulations outside the modern differentiable-programming sense

The discriminative Kalman filter is a separate, well-defined use of the acronym. It retains linear-Gaussian latent dynamics but replaces the generative observation model with a direct Gaussian approximation to the state posterior,

$S_n = H_n P_{n|n-1} H_n^T + R_n,$1

With predictive covariance

$S_n = H_n P_{n|n-1} H_n^T + R_n,$2

the closed-form recursion becomes

$S_n = H_n P_{n|n-1} H_n^T + R_n,$3

$S_n = H_n P_{n|n-1} H_n^T + R_n,$4

This formulation performed substantially better than KF, EKF, and UKF on two synthetic nonlinear/non-Gaussian benchmarks and on rhesus macaque neural decoding, where normalized MSE averages were reported as $S_n = H_n P_{n|n-1} H_n^T + R_n,$5 for DKF-GP, $S_n = H_n P_{n|n-1} H_n^T + R_n,$6 for DKF-GP-freq, and $S_n = H_n P_{n|n-1} H_n^T + R_n,$7 for DKF-NN, versus $S_n = H_n P_{n|n-1} H_n^T + R_n,$8 for the Kalman filter (Burkhart et al., 2016). A later human-neural-decoding study used the same acronym for a discriminative approach to Bayesian filtering and reported successful real-time use within the BrainGate2 clinical trial, including cursor control for three volunteers with quadriplegia and tablet typing by participant “T9” (Burkhart, 2018).

The desensitized Kalman filter is another distinct use. For a linear discrete-time system with uncertain constant parameter vector $S_n = H_n P_{n|n-1} H_n^T + R_n,$9,

$K_n = P_{n|n-1} H_n^T S_n^{-1},$0

the paper defines sensitivity matrices

$K_n = P_{n|n-1} H_n^T S_n^{-1},$1

and minimizes

$K_n = P_{n|n-1} H_n^T S_n^{-1},$2

The resulting analytical gain is

$K_n = P_{n|n-1} H_n^T S_n^{-1},$3

A central result is that when the sensitivity-weighting matrix is chosen as the parameter covariance,

$K_n = P_{n|n-1} H_n^T S_n^{-1},$4

the special desensitized Kalman filter is mathematically equivalent to the consider Kalman filter, with the bridge

$K_n = P_{n|n-1} H_n^T S_n^{-1},$5

This paper resolves the DKF weighting choice by setting it equal to the a priori uncertain-parameter covariance (Lou, 2015).

The phrase “deep Kalman filter” is also not identical to “differentiable Kalman filter.” In the theoretical work on continuous-time DKFs, the model is a neural architecture

$K_n = P_{n|n-1} H_n^T S_n^{-1},$6

that outputs Gaussian measures and is proved to uniformly approximate the conditional law of a broad class of non-Markovian, conditionally Gaussian signal processes on regular compact path sets. The approximation guarantee is stated in Wasserstein distance: $K_n = P_{n|n-1} H_n^T S_n^{-1},$7 This provides a theoretical tie between deep Gaussian-output sequence models and classical stochastic filtering, but it addresses a broader functional-approximation question than the usual end-to-end backpropagation-through-Kalman-recursion setting (Horvath et al., 2023).

5. Geometry, manifolds, and what “differentiable” can also mean

In another neighboring literature, “differentiable” refers not to trainability but to state spaces that are differentiable manifolds. The on-manifold Kalman-filter framework defines

$K_n = P_{n|n-1} H_n^T S_n^{-1},$8

together with an $K_n = P_{n|n-1} H_n^T S_n^{-1},$9 operation for system evolution, and writes the canonical on-manifold system as

$\hat x_{n|n} = \hat x_{n|n-1} + K_n z_n,$0

This yields a generic symbolic framework for prediction, update, and full iterated Kalman filtering on manifolds composed of $\hat x_{n|n} = \hat x_{n|n-1} + K_n z_n,$1, $\hat x_{n|n} = \hat x_{n|n-1} + K_n z_n,$2, and $\hat x_{n|n} = \hat x_{n|n-1} + K_n z_n,$3, implemented in the IKFoM toolkit (He et al., 2021).

This manifold perspective is orthogonal to the modern DKF notion. It uses Jacobians and smooth structure, but it is not an autodiff-based learning framework. The distinction matters because many robotic filtering systems combine both ideas in practice: state representations may be on manifolds, while learned measurement or uncertainty models are trained through differentiable updates. The robocentric visual-inertial DKF is an example of such a hybridization, since it uses an EKF state involving orientations, gravity, velocity, and biases while still backpropagating self-supervised losses through the filtering pipeline (Wagstaff et al., 2022).

6. Limitations, misconceptions, and current directions

A common misconception is that “DKF” always means differentiable Kalman filter. The literature does not support that reading: discriminative, desensitized, deep, distributed, discrete, and differentiable-manifold usages are all active, and several recent distributed-optimization papers explicitly distinguish their terminology from the machine-learning sense (Iqbal et al., 22 Jan 2025, Yang et al., 11 Apr 2025).

Another misconception is that differentiability implies that the entire filter must be learned. The cited implementations suggest the opposite. In visual-inertial odometry, the IMU-driven process model is retained and only the visual measurement model and its covariance are learned; in cooperative tracking, the constant-velocity model and process covariance are fixed while measurement uncertainty is learned; in online adaptive filtering, the predictive measurement distribution is adapted through an OT loss rather than replacing the filter with a black-box recurrent model (Wagstaff et al., 2022, Chiu et al., 2023, He et al., 9 Aug 2025). This suggests that the dominant role of DKF methods is to expose selected components of the filtering pipeline to gradient-based optimization, not to discard model structure.

The current literature also indicates several persistent technical constraints. Closed-form backpropagation results are derived for the classical linear-Gaussian Kalman filter and a scalar loss, even though they are highly useful as differentiable primitives (Parellier et al., 2023). Practical applications often restrict covariances to diagonal or block-diagonal forms for stability and tractability, as in learned per-detection observation covariances for cooperative tracking and diagonal visual measurement covariances for self-supervised VIO (Chiu et al., 2023, Wagstaff et al., 2022). When the latent system is high-dimensional, full covariance propagation becomes expensive; DKFNet explicitly points to block-diagonal, localized, low-rank, PCA, or SVD approximations as scalability strategies (Wu et al., 9 Sep 2025). At the theoretical end, universal approximation results for deep Kalman filters hold uniformly only on sufficiently regular compact subsets of path space, not on all of infinite-dimensional path space (Horvath et al., 2023).

Taken together, these results position the differentiable Kalman filter not as a single algorithmic object but as a research program: preserve the recursive probabilistic structure of Kalman filtering, expose the relevant algebra to differentiation, and learn the parts of the model that are uncertain, misspecified, or nonstationary.