Learning Performance-oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation

For a safe set described by Boolean logical operations on multiple constraints, [6] composes multiple CBFs accordingly through the non-smooth min/max operators.[7] introduces smooth composition of logical operations on constraints, which was later extended to simultaneously handle actuation and state constraints [8][9] using integral CBFs [12]. [15] proposes an algorithmic way to create a single smooth CBF arbitrarily composing both min and max operators. Such smooth bounds have been used in changing environments [16]. Notably, [17] ensures the input constrained feasibility of the CBF condition while composing multiple CBFs.

High-order CBF (HOCBF) [10] and exponential CBF [11] are systematicly approach CBF construction when the safety constraints have high relative degree. However, controlling the conservatism of these approaches is a challenge. To reduce conservatism or improve the performance of HOCBF, various learning frameworks have been proposed [18, 19, 20] that allow tuning of the class $\mathcal{K}$ functions used in the CBF condition.

Motivated by the difficulty of hand-designing CBFs, learning-based approaches building on the past [21] have emerged in recent years as an alternative [22, 23, 24, 25]. Liu et al. [26] explicitly consider input constraints in learning a CBF by finding counterexamples on the $0$-level set of the CBF for training, while Dai et al. [27] propose a data-efficient prioritized sampling method. [28] explores adaptive sampling for training HJB NN models favoring sharp gradient regions. Drawing tools from reachability analysis, the recent work [29] iteratively expands the volume of the control invariant set by learning the policy value function and improving the performance through policy iteration. Similarly, Dai et al. [30] expand conservative hand-crafted CBFs by learning on unsafe and safe trajectories, and Qin et al. [31] applies actor-critic RL framework to learn a CBF value function while [32] learns a control policy and CBF together for black-box constraints and dynamics. Our method differs by being controller-independent and basing its learning objective on the HJ partial differential equation (PDE), which defines the maximal control invariant set, without requiring trajectory training data.

The value functions in HJ reachability analysis have been extended to construct control barrier-value functions (CBVF) [33] and control Lyapunov-value functions [34], which can be computed using existing toolboxes [35]. Tokens et al. [36] further apply such tools to refine existing CBF candidates, and Bansal et al. [37] learn neural networks solution to a HJ PDE for reachability analysis. Of particular interest to our work is the CBVF, which is close to the CBF formulation and provides a characterization of the viability kernel. Our work aims to learn a neural network CBF from data without using computational tools based on spatial discretization.

An extended class $\mathcal{K}$ function is a function $\alpha:(-b,a)\mapsto\mathbb{R}$ for some $a,b>0$ that is strictly increasing and satisfies $\alpha(0)=0$. We denote $L_{r}^{+}(h)=\{x\in\mathbb{R}^{n}\mid h(x)\geq r\}$ as the $r$-superlevel set of a continuous function $h(x)$. The positive and negative parts of a number $a\in\mathbb{R}$ are denoted by $(a)_{+}=\max(a,0)$ and $(a)_{-}=\min(a,0)$, respectively. Let $\land,\lor,\neg$ denote the logical operations of conjunction, disjunction, and negation, respectively. For two statements $A$ and $B$, we have $\neg(A\land B)=(\neg A)\lor(\neg B)$ and $\neg(A\lor B)=(\neg A)\land(\neg B)$.

For $t\leq 0$, we have $\mathcal{V}(t)=\{x\mid B_{\gamma}(x,t)\geq 0\}$ [33, Proposition 2]. Additionally, $B_{\gamma}$ is the unique Lipschitz continuous viscosity solution of the following HJ PDE with terminal condition $B_{\gamma}(x,0)=\ell(x)$ [33, Theorem 3]:

Under mild assumptions of Lipschitz continuous dynamics and $\ell(x)$ being a signed distance function, $B_{\gamma}(x,t)$ is differentiable almost everywhere [39]. Furthermore, taking $t\rightarrow-\infty$, the steady state solution $B_{\gamma}(x):=B_{\gamma}(x,-\infty)$ gives us the maximal control invariant set contained in $L_{0}^{+}(\ell)$ [33, Section II.B]:

We will leverage the above PDE to learn a CBF $h$ whose zero superlevel set approximates the maximal control invariant set.

CBF parameterization In the context of our problem, we are interested in the viability kernel of $L_{0}^{+}(\underline{c}(x))$, where $\underline{c}$ is the smoothed composition of the constraints. Thus, our goal is to learn a CBF $h$ that satisfies the PDE $\mathcal{N}(h,\underline{c})=0$. To this end, we parameterize the CBF candidate as

where $\delta_{\theta}\colon\mathcal{D}\to\mathbb{R}_{\geq 0}$ is a non-negative continuously differentiable function approximator with parameters $\theta$. In this paper, we parameterize $\delta_{\theta}$ in the form of a multi-layer perceptron:

where $\{W_{k},b_{k}\}_{k=0}^{K}$ are the learnable parameters, $\sigma(\cdot)$ is a smooth activation such as the ELU, Tanh, or Swish functions [40], and $\sigma_{+}(\cdot)$ is a smooth function with non-negative outputs, i.e., $\sigma_{+}(r)\geq 0,\forall r\in\mathbb{R}$. We choose $\sigma_{+}(\cdot)$ as the Softplus function, $\sigma_{+}(r)=\frac{1}{\beta}\log(1+\exp(\beta r)),\;\beta>0$.

The parameterization (9) offers several advantages over using a standard MLP model for $h_{\theta}$. First, it accelerates learning by enforcing the non-negativity constraint $\underline{c}(x)\geq h(x)$ in (8) by design. Second, complex constraints are directly integrated into the system via $\underline{c}(x)$, simplifying the learning process. Importantly, $h_{\theta}(x)$ is automatically non-positive in regions containing obstacles, allowing the learning process to focus elsewhere.

In summary, the parameterization in (9) ensures that $h_{\theta}(\cdot)$ is smooth and satisfies $h_{\theta}(x)\leq\underline{c}(x)\leq c(x)$ for all $x\in\mathcal{D}$, implying that $L_{0}^{+}(h_{\theta})\subseteq L_{0}^{+}(\underline{c})\subseteq L_{0}^{+}({c})=\mathcal{X}$ by construction.

Initialization We propose a specialized initialization scheme by letting $W_{L},b_{L}=0$, initializing the candidate $h_{\theta}$ with $\underline{c}$. This initialization favors finding the closest CBF to $\underline{c}(x)$ during training. Setting only the last layer to zero allows training to break symmetry and retain gradient flow.

Training loss and sampling distribution Given the parameterization (9), we now propose to train $h_{\theta}$ to approximately satisfy the steady-state HJ PDE. This leads to the following physics-informed risk minimization

The first loss $\mathcal{L}_{\text{HJ}}(x;\theta,\gamma)$ is the square of the PDE residual to increase the volume of $L_{0}^{+}(h_{\theta})$, while the second loss $\mathcal{L}_{\text{CBF}}(x;\theta,\gamma)$ enforces the CBF condition. We note that the latter is implicitly present in the PDE but not enforced by the corresponding loss $\mathcal{L}_{\text{HJ}}(x;\theta,\gamma)$.

Finally, $\mu$ represents a sampling distribution over $L_{0}^{+}(\underline{c})$, meaning no learning is required within obstacle regions. Efficient sampling in these regions can be achieved using intelligent sampling methods like envelope rejection [41] or the random walk Metropolis-Hastings (RWMH) [42], both of which are well-suited for cluttered environments. Notably, the training process remains self-supervised, as only domain points are collected for learning.

In the following, we demonstrate that jointly enforcing $\mathcal{L}_{\text{HJ}}$ and $\mathcal{L}_{\text{CBF}}$ together with intelligent sampling (IS) improves the performance and safety of the learned PINN-CBF. We note that for polyhedral actuation constraint sets, the supremum over $u$ is achieved at one of the vertices, yielding a closed-form expression. We summarize our self-supervised training method in Algorithm 1.

First, we consider the double integrator benchmark $\dot{x}=v\quad\dot{v}=u$, with $x\in\mathbb{R}$ denoting the position and $v\in\mathbb{R}$ denoting the velocity. The action $u\in\mathbb{R}$ represents the acceleration of the system and directly operates on $v$.

We generate a complex obstacle configuration consisting of rotated rectangles and unit walls bordering the state space $\mathcal{D}=\{(x,v)\mid-1\leq x\leq 11,-6\leq v\leq 6\}$ shown in Fig. 2. Then, we parameterize PINN-CBF following Section 3.3 and train it with $N=10^{4}$ uniform samples. At each state in the training data, we compute the values and gradient of $\underline{c}$ using JAX [43] flexible automatic-differentiation and hardware acceleration. The acceleration $u$ is bounded by $u\in[-1,1]$. The nominal controller is given by a PID controller meant to stabilize the system to a target position (see Section A.1).

Fixed-Wing Aircraft The Dubins fixed-wing Plane system has 7D states and 3D limited actuation control. Its full dynamics are shown in Appendix A.2. Sampling and training are done as before except with $N=10^{6}$ to compensate for a higher dimensional space $\mathcal{D}=\{(n,e,d,\phi,\theta,\psi,V_{T})\mid-6\leq n,e\leq 6,-1\leq d\leq 11,-2\pi\leq\phi,\theta,\psi\leq 2\pi,0.5\leq V_{T}\leq 2\}$. Although $10^{6}\ll(\sqrt{10^{4}})^{7}=10^{14}$, as suggested by naive dimensional scaling, we aim to demonstrate how learning can overcome the curse of dimensionality. The action $u=\{A_{T},p,q\}$ is bounded in magnitude by $|u|\leq[10.5,1,1]$. Rectangular obstacles occur in $n,e,d$. We chose a nominal trajectory resembling takeoff to evaluate the PINN-CBF filter against relevant baselines on collision avoidance.

We evaluate the combination of loss functions (11b) and (11c) with intelligent sampling (IS) from Algorithm 1, as shown in Table 1. All proposed ablations and baseline methods use the same architecture (9) for a fair comparison, accommodating complex input-space constraints. The MLP for $\delta_{\theta}$ has layer widths of $2-50-50-1$, while DeepReach [37] includes time as an additional input and is applied over the full horizon with $t=0$.

In the ablations, scalarization with $\lambda=0.2$ reduces safety violations but decreases volume. In contrast, intelligent sampling lowers residuals and safety violations while increasing volume, suggesting it mitigates the tradeoff of scalarization. Among the baselines, directly enforcing the CBF condition (2) with $\mathcal{L}_{\text{CBF}}^{2}$ results in a highly safe but ineffective filter with zero volume. NCBF and DeepReach achieve high volume but with significantly higher safety errors. Next, we will examine how these methods perform on a more complex fixed-wing plane system.

Fixed-Wing Plane The same comparisons conducted for the Double-Integrator are also performed with the following adjustments: all methods utilize a $7-100-100-1$ MLP architecture. DeepReach includes an additional input for time. Additionally, validation is carried out on $10^{7}$ uniformly random sampled states instead of grid samples.

Training with $\mathcal{L}_{\text{CBF}}^{2}$ is ineffective, yielding nearly zero volume. The performance of NCBF severely declines due to the increased difficulty of sampling for counterexamples in higher dimensions. In contrast, DeepReach achieves higher volume at the cost of increased safety violations. Our method, PINN-CBF, achieves both low conservatism and safety. To illustrate this, we visualize the performance of PINN-CBF, NCBF, and DeepReach in filtering a nominal trajectory representing takeoff.

Fixed-Wing Aircraft Example In Fig. 4, the fixed-wing aircraft starting from the given position collides with a 3D obstacle, while in Fig. 4, obstacles are avoided with the PINN-CBF safety filter.