SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this section, we provide the necessary definitions, lemmas, and remarks that are required for introducing and addressing the CI propagation problem in Section 4.1. That is, how to propagate, with probabilistic guarantees, a CI constructed at the locations of the available data to arbitrary sets of query points of the domain.
To this aim, we first define the marginal CI for a single query point  $x_{q}$ .
Definition E.1 (Marginal CI). A pair of random variables  $(L, U) \in \mathbb{R}^2$  is a  $p$ -CI for  $u(x_q)$  on the single query point  $x_q \in \Omega$ , if the random event  $\mathcal{E} := \{L < u(x_q) < U\}$  holds with probability at least  $p$ .
Intuitively, considering repeating experiments of drawing random datasets  $\{z^{(j)}\}_{1:m}$ , the proportion of the  $(L,U)$  CIs, constructed according to Definition E.1, that contain  $u(x_{q})$  is at least  $p$ . Next, we define the joint CI for multiple query points.
Definition E.2 (Joint CI). A pair of functions  $(L, U) \in \mathbb{R}^{\mathcal{X}_q} \times \mathbb{R}^{\mathcal{X}_q}$  is a  $p$ -CI for  $u$  on query set  $\mathcal{X}_q \subseteq \Omega$ , if the random event  $\mathcal{E} := \cap_{x_q \in \mathcal{X}_q} \{L(x_q) < u(x_q) < U(x_q)\}$  holds with probability at least  $p$ .
For simplicity, in the rest of the paper we refer to the  $p$ -CI for  $u$  on the whole domain  $\Omega$  as  $p$ -CI for  $u$ , and to the  $p$ -CI for  $u$  on  $\mathcal{X}_q \subset \Omega$  as query set  $p$ -CI.
Remark E.3. Let  $(L,U)$  be a  $p$ -CI for  $u$ . For any  $\mathcal{X}_q \subseteq \Omega$ ,  $(L,U)$  is a  $p$ -CI for  $u$  on  $\mathcal{X}_q$ , i.e., the random event  $\mathcal{E}(\mathcal{X}_q,Z) \coloneqq \cap_{x_q \in \mathcal{X}_q} \{L(x_q) < u(x_q) < U(x_q)\}$  holds with probability at least  $p$ .
Remark E.3 indicates that a  $p$ -CI for  $u$  on an arbitrary query set can be derived from a  $p$ -CI for  $u$  by slicing. The query set  $\mathcal{X}_q$  can be either discrete or continuous including multiple or even infinite queries. Note that the  $p$ -CIs of Definition E.2 and Remark E.3 hold with probability at least  $p$ ; i.e., there is no statement about their tightness. In this regard, the next remark shows that slicing using larger sets  $\mathcal{X}_q$  yields tighter  $p$ -CIs for  $u$  on  $\mathcal{X}_q$ . Nevertheless, the maximum tightness that can be achieved, for the  $p$ -CI for  $u$  on  $\mathcal{X}_q$ , is determined by the tightness of the  $p$ -CI for  $u$  on the whole domain  $\mathcal{X}_q$ . This is corroborated by the following remark.
Remark E.4. Let  $(L,U)\in \mathbb{R}^{\Omega}\times \mathbb{R}^{\Omega}$  be a  $p$ -CI for  $u$ . For query set  $\mathcal{X}_q^1\subseteq \mathcal{X}_q^2\subseteq \Omega$ , the probabilities of random events  $\mathcal{E}_1\coloneqq \cap_{x_q\in \mathcal{X}_q^1}\{L(x_q) < u(x_q) < U(x_q)\}$ ,  $\mathcal{E}_2\coloneqq \cap_{x_q\in \mathcal{X}_q^2}\{L(x_q) < u(x_q) < U(x_q)\}$  satisfy
$$
\Pr \left(\mathcal {E} _ {1} \geq \Pr \left(\mathcal {E} _ {2}\right). \right. \tag {16}
$$
Next, we define the union of single-query  $p$ -CIs. It is often used in practice as a proxy to the query set  $p$ -CI and can be used also within the context of the present study for addressing problem scenarios 3 and 4, as described in Appendix D.
Definition E.5 (Union of CIs). A pair of functions  $(L,U)\in \mathbb{R}^{\mathcal{X}_q}\times \mathbb{R}^{\mathcal{X}_q}$  constructed based on  $\{z^{(j)}\}_{1:m}$  is a union of  $p$  -CIs for  $u$  on query set  $\mathcal{X}_q\subseteq \Omega$  if for each  $x_{q}\in \mathcal{X}_{q}$ $(L(x_{q}),U(x_{q}))$  is a  $p$  -CI for  $u(x_q)$  , i.e., each random event  $\mathcal{E}:= \{L(x_q) < u(x_q) < U(x_q)\}$  holds with at least probability  $p$
The next lemma provides insight into the connection between the marginal  $p$ -CI on  $\mathcal{X}_q$  and the union of  $p$ -CIs.
Lemma E.6. For  $u$  and  $\mathcal{X}_q$ , a  $p$ -CI is a union of  $p$ -CIs, whereas a union of  $p$ -CIs is not necessarily a  $p$ -CI.
That is,  $u(x_{q}), \forall x_{q} \in \mathcal{X}_{q}$ , belongs to the  $p$ -CI constructed by slicing of the  $p$ -CI for  $u$  with probability at least  $p$ . On the other hand, the probability of the event  $\cap_{x_q \in \mathcal{X}_q} \{L(x_q) < u(x_q) < U(x_q)\}$ , where  $(L, U)$  are constructed based on a union of  $p$ -CIs, may be less than  $p$ . This is supported by the following two remarks as well as the computational experiment of Appendix F.
Remark E.7. The probability of the random event  $\cap_{x_q\in \mathcal{X}_q}\{L(x_q) < u(x_q) < U(x_q)\}$ , where  $(L,U)$  are constructed based on a union of  $p$ -CIs, is upper-bounded by the minimum probability of the events  $\{L(x_q) < u(x_q) < U(x_q)\}$ ,  $\forall x_{q}\in \mathcal{X}_{q}$ . That is,
$$
\Pr \left(\cap_ {x _ {q} \in x _ {q}} \left\{L \left(x _ {q}\right) <   u \left(x _ {q}\right) <   U \left(x _ {q}\right) \right\}\right)
$$
$$
\leq \min  _ {x _ {q} \in \mathcal {X} _ {q}} \Pr \left(\left\{L \left(x _ {q}\right) <   u \left(x _ {q}\right) <   U \left(x _ {q}\right) \right\}\right). \tag {17}
$$
Remark E.8. If one of the  $p$ -CIs is tight, i.e.  $\exists x_{q}^{\prime}\in \mathcal{X}_{q}$  such that  $\operatorname*{Pr}(\{L(x_q') < u(x_q') < U(x_q')\}) = p$ , the union of these  $p$ -CIs is a  $p$ -CI for  $u$  if and only if it holds  $\forall x_{q}\in \mathcal{X}_{q}\setminus \{x_{q}^{\prime}\}$  that
$$
\Pr \left(\left\{L \left(x _ {q}\right) <   u \left(x _ {q}\right) <   U \left(x _ {q}\right) \mid L \left(x _ {q} ^ {\prime}\right) <   u \left(x _ {q} ^ {\prime}\right) <   U \left(x _ {q} ^ {\prime}\right) \right\}\right) = 1. \tag {18}
$$
Finally, in order to obtain a guarantee that a union of CIs is a  $p$ -CI, we can equate a lower bound of the probability of the former with  $p$ . This lower bound can be obtained by disregarding correlations among the different locations, as formalized in the following lemma.
Lemma E.9. For  $u$  and finite query set  $\mathcal{X}_q$ , a union of  $(1 - (1 - p) / |\mathcal{X}_q|)$ -CIs is a  $p$ -CI for  $u$  as
$$
\begin{array}{l} \Pr \left(\cap_ {x _ {q} \in \mathcal {X} _ {q}} \left\{L \left(x _ {q}\right) \leq u \left(x _ {q}\right) \leq U \left(x _ {q}\right) \right\}\right) \\ = 1 - \Pr \left(\cup_ {x \in \mathcal {X} _ {q}} \{L (x) \leq f (x) \leq U (x) \} ^ {C}\right) \\ \geq 1 - \sum_ {x \in \mathcal {X} _ {q}} (1 - \Pr (\{L (x) \leq u (x) \leq U (x) \})) \tag {19} \\ \geq 1 - | \mathcal {X} _ {q} | (1 - (1 - \frac {1 - p}{| \mathcal {X} _ {q} |})) = p. \\ \end{array}
$$
However, as the size of the query set  $\mathcal{X}_q$  increases, the required confidence level of each interval must approach 1, which leads to loose union bounds. Finally, another approach for obtaining a  $p$ -CI for multiple queries is to use multivariate concentration inequalities [8]. Nevertheless, this approach is limited to a finite number of queries.
F Toy Linear System
In this section, we highlight the differences between joint CIs and union of CIs, as described in Appendix E, with a simple example. Consider a true function  $u$  and a model given as  $u_{\theta}(x) = (1 - x)\theta_0 + x\theta_1$ ,  $x \in [0,1]$ , with  $\theta = (\theta_0,\theta_1) \in \mathbb{R}^2$ . We assume that there exists  $\theta^{*}$  such that  $u_{\theta^{*}} = u$ , and that we are given measurements  $Z = (Z_{0},Z_{1}) \sim \mathcal{N}(\theta^{*},I)$ . Thus, for one observation  $Z$ , we simply estimate  $\hat{\theta} = Z$ . As a result, the estimator for  $u$  is given as
$$
u _ {\hat {\theta}} = (1 - x) Z _ {0} + x Z _ {1} \sim \mathcal {N} \left(\left(1 - x\right) \theta_ {0} ^ {*} + x \theta_ {1} ^ {*}, 2 x ^ {2} - 2 x + 1\right). \tag {20}
$$
Next, define as  $k$ -interval the lower and upper intervals  $(L_k, U_k)$  obtained by subtracting and adding  $k$  standard deviations from the estimated mean of  $u$ , respectively, i.e.,
$$
L _ {k} (x) := (1 - x) Z _ {0} + x Z _ {1} - k \sqrt {2 x ^ {2} - 2 x + 1}, \tag {21}
$$
$$
U _ {k} (x) := (1 - x) Z _ {0} + x Z _ {1} + k \sqrt {2 x ^ {2} - 2 x + 1}.
$$
Remark F.1. We define  $k_p^{\mathcal{N}}$ -interval as the interval constructed as a union of one-query  $p$ -CIs for  $u_{\theta}(x)$  and for different values of  $x \in \Omega$ , similarly to Definition E.5. The value  $k_p^{\mathcal{N}}$  is the upper  $(1 - p) / 2$  critical value of the standard normal distribution.
Proof: For  $x\in \Omega$
$$
\begin{array}{l} \Pr \left(\left\{L _ {k _ {p} ^ {\mathcal {N}}} \leq u (x) \leq U _ {k _ {p} ^ {\mathcal {N}}} \right\}\right) \\ = \Pr \left(\left\{(1 - x) Z _ {0} + x Z _ {1} - k _ {p} ^ {\mathcal {N}} \sqrt {2 x ^ {2} - 2 x + 1} <   f (x) <   (1 - x) Z _ {0} + x Z _ {1} + k _ {p} ^ {\mathcal {N}} \sqrt {2 x ^ {2} - 2 x + 1} \right\}\right. \tag {22} \\ = \Pr \left(\left\{\left| u _ {\hat {\theta}} (x) - \mathbb {E} \left[ u _ {\hat {\theta}} (x) \right] \right| <   k _ {p} ^ {\mathcal {N}} \operatorname {s t d} \left(u _ {\hat {\theta}} (x)\right) \right\}\right) \\ = p. \\ \end{array}
$$
Remark F.2. We define  $k_p^{\chi^2}$ -interval as the  $p$ -CI for  $u$ , constructed as  $(L_{k_p^{\chi^2}}(x), U_{k_p^{\chi^2}}(x))$  for each  $x \in \Omega$ , where
$$
L _ {k _ {p} ^ {\chi^ {2}}} (x) = \underset {| | z - Z | | _ {2} \leq k _ {p} ^ {\chi^ {2}}} {\arg \min } (1 - x) z _ {0} + x z _ {1}, \tag {23}
$$
$$
U _ {k _ {p} ^ {\chi^ {2}}} (x) = \underset {| | z - Z | | _ {2} \leq k _ {p} ^ {\chi^ {2}}} {\arg \max } (1 - x) z _ {0} + x z _ {1}. \tag {23}
$$
The value  $k_{p}^{x^{2}}$  is the squared root of the upper  $(1 - p) / 2$  critical value of the chi-squared distribution with 2 degrees of freedom.
Proof: According to Theorem 4.1,
$$
\begin{array}{l} \Pr (\{L _ {k _ {p} ^ {\chi^ {2}}} (x) <   u (x) <   U _ {k _ {p} ^ {\chi^ {2}} (x)}, \forall x \in \Omega \}) \\ \geq \Pr \left(\left\{\left| \left| \theta^ {*} - Z \right| \right| _ {2} \leq k _ {p} ^ {\chi^ {2}} \right\}\right) \tag {24} \\ = p. \\ \end{array}
$$
The left side of Figure F.1 provides an indicative example of a  $k_{p}^{\mathcal{N}}$ -interval and a  $k_{p}^{x^2}$ -interval, with  $p = 95\%$  and  $\theta^{*} = (0,0)$ . The results are obtained using only one sample  $Z = (Z_0,Z_1)$ , which can be visualized by considering that  $u_{\theta}(0) = Z_0$  and  $u_{\theta}(1) = Z_1$ . Further, the empirical confidence level of an interval  $(L,U)$  can be estimated as  $\bar{p} := \frac{1}{J}\sum_{j=1}^{J}\mathbf{1}(\cap_{k=0,1,\dots,\lfloor 1/\epsilon\rfloor}\{LZ_j(k\epsilon)<u(k\epsilon)<UZ_j(k\epsilon)\})$  with  $\epsilon > 0$ . That is, we calculate the proportion of times that such a construction of the interval contains the true values on  $\{0,\epsilon,\dots,\lfloor 1/\epsilon\rfloor\epsilon\}$ . According to the law of large numbers, the empirical confidence level converges with increasing  $J$ , in probability, to the true confidence level  $p$ . The right side of Figure F.1 shows that the empirical confidence level curve of the  $k_{p}^{x^2}$ -interval (or the  $k_{p}^{\mathcal{N}}$ -interval) converges to a value above (or below)  $95\%$ . This result implies that the  $k_{p}^{x^2}$ -interval is a valid  $95\%$ -CI for  $u$  while the  $k_{p}^{\mathcal{N}}$ -interval is not.
Figure F.1: Left: An indicative example of a  $k_p^{\mathcal{N}}$ -interval and a  $k_p^{x^2}$ -interval, with  $p = 95\%$  and  $\theta^* = (0,0)$ . The results are obtained using only one sample  $Z = (Z_0,Z_1)$ , which can be visualized by considering that  $u_{\theta}(0) = Z_0$  and  $u_{\theta}(1) = Z_1$ . Right: Empirical confidence level on  $\Omega$  versus the number of samples  $J$ .
G.1 Proof of Theorem 4.1
If  $\bar{z} \in \tilde{\mathcal{Z}}_p$ , then  $L(x) = \min_{z \in \tilde{\mathcal{Z}}_p} u(x; \bar{z}) \leq u_A(x; \bar{z}) \leq \max_{z \in \tilde{\mathcal{Z}}_p} u(x; \bar{z}) = U(x)$ , for all  $x \in \Omega$ . Further, according to the definition of the proper solver, we have  $u_A(x; \bar{z}) - \eta \leq u(x) \leq u_A(x; \bar{z}) + \eta$  for all  $x \in \Omega$ , and thus  $L(x) - \eta \leq u(x) \leq U(x) + \eta$ , for all  $x \in \Omega$ . Finally, we can conclude that  $Pr\{L(x) - \eta \leq u(x) \leq U(x) + \eta, \forall x \in \Omega\} \geq Pr\{\bar{z} \in \tilde{\mathcal{Z}}_p\} \geq p$ .
H Algorithms
The developed methods are summarized in Algorithms 1 and 2.
Algorithm 1 PICProp: Physics-Informed Confidence Propagation (Gradient-based)
Input:  $\tilde{\mathcal{Z}}_p(p\text{-}\mathrm{CI}\text{for}\tilde{z})$  query point  $x$  inner-steps schedule  $\{N_k\}_{k = 1}^{K_o}$  , learning rates  $\alpha ,\beta$  , outer steps  $K_{o}$    
Initialize  $z_0\in \tilde{\mathcal{Z}}_p$ $\theta_0$    
for  $k = 1,\dots,K_{o}$  do for  $n = 1,\ldots ,N_{k}$  do  $\theta_{n} = \theta_{n - 1} - \alpha \nabla_{\theta}\mathcal{L}_{pinn}(\theta_{n - 1},z_{k - 1})$  end for  $\hat{\theta} = \theta_0 = \theta_{N_k}$  if lower bound is sought then  $z_{k} = z_{k - 1} - \beta \nabla_{z}u_{\hat{\theta} (z_{k - 1})}(x)$  else  $z_{k} = z_{k - 1} + \beta \nabla_{z}u_{\hat{\theta} (z_{k - 1})}(x)$  end if   
end for   
return  $u_{\hat{\theta} (z_{K_o})}(x)$
I Bi-level Optimization
A bi-level optimization problem can be formulated as
$$
\min  _ {\phi} \mathcal {J} \left(\theta^ {*} (\phi), \phi\right) \quad s. t. \theta^ {*} (\phi) = \underset {\theta} {\arg \min } \mathcal {L} (\theta , \phi), \tag {25}
$$
Algorithm 2 EffiPICProp: Efficient Physics-Informed Confidence Propagation
Input:  $\eta ,p(x_q),\lambda$    
Randomly sample  $x_{q}^{1:K}\sim p(x_{q})$    
Obtain  $\hat{\theta} (z^L (x_q^k)),\hat{\theta} (z^U (x_q^k))$  with Algorithm 1 for  $k = 1,\dots,K$  Initialize  $\psi$    
while not done do  $\psi \gets \psi -\eta \nabla \mathcal{L}(x_q^{1:K};\psi)$  according to Equation (7).   
end while   
return  $u_{\psi}$
where  $\theta, \phi$  are the parameters and the hyperparameters, whereas  $\mathcal{L}$  and  $\mathcal{J}$  are the lower-level objective and the upper-level objective. Many problems like meta-learning, hyperparameter optimization, data poisoning, and auxiliary learning can be naturally formulated in a bi-level optimization way.
The hardest part of bi-level optimization is that  $\theta^{*}(\phi)$  is not available in a closed-form in many scenarios, particularly with an empirical lower-level objective that requires multiple steps of gradient descent. In such cases, the explicit gradient of high-level objective with respect to hyperparameter is intractable.
According to the chain rule,
$$
\nabla_ {\phi} \mathcal {J} \left(\theta^ {*} (\phi), \phi\right) = \nabla_ {\theta} \mathcal {J} \nabla_ {\phi} \theta^ {*} (\phi) + \nabla_ {\phi} \mathcal {J}. \tag {26}
$$
The most challenging part is the approximation of the implicit gradient  $\nabla_{\phi}\theta^{*}(\phi)$ , for which many advanced methods have been proposed and studied in recent years [15]. In this paper, three methods referred to as Reverse, approximate implicit differentiation with Neumann Series (AID-NS), and approximate implicit differentiation with conjugate gradient (AID-CG) are evaluated on the confidence interval identification tasks.
I.1 Reverse
Reverse is an iterative differentiation-based method to approximate the hypergradient by unrolling the inner optimization  $\theta_{1:K}$  trajectory in a reverse way. The implicit gradient can be approximated by
$$
\nabla_ {\phi} \theta^ {*} (\phi) \approx \nabla_ {\phi} \theta_ {K} (\phi) \approx \sum_ {k = 1} ^ {K - 1} \left(\prod_ {i = k + 1} ^ {K - 1} \frac {\partial \tilde {\theta} _ {i + 1}}{\partial \theta_ {i}}\right) \frac {\partial \tilde {\theta} _ {k + 1}}{\phi}, \tag {27}
$$
where  $\tilde{\theta}_{k + 1} = \theta_{k} - \alpha \nabla_{\theta}l(\theta_{k},\phi)$  and  $\alpha$  is a pre-defined inner-learning rate.
I.2 Approximate Implicit Differentiation (AID)
Another family of methods to approximate the hypergradient is Implicit Differentiation, where the implicit function theorem (IFT) is used to evaluate  $\nabla_{\phi}\theta^{*}(\phi)$ . If  $\mathcal{L}$  is second-order derivable and strong convex, we have
$$
\nabla_ {\phi} \theta^ {*} (\phi) = - \left(\nabla_ {\theta} ^ {2} \mathcal {L}\right) ^ {- 1} \nabla_ {\phi} \nabla_ {\theta} \mathcal {L}. \tag {28}
$$
Then we can approximate the gradients of  $\phi$  using
$$
\nabla_ {\phi} \mathcal {J} \left(\theta^ {*} (\phi), \phi\right) = - \nabla_ {\theta} \mathcal {J} \cdot \left(\nabla_ {\theta} ^ {2} \mathcal {L}\right) ^ {- 1} \nabla_ {\phi} \nabla_ {\theta} \mathcal {L} + \nabla_ {\phi} \mathcal {J}. \tag {29}
$$
The most challenging part is the approximation of the inverse Hessian vector product (inv-hvp)  $A^{-1}p$ , where  $A \in \mathbb{R}^{d \times d}$  is a given symmetric positive matrix and  $p \in \mathbb{R}$  is a given vector.
AID-NS [19] is a method based on the Neumann Series to approximate the inv-hvp as
$$
A ^ {- 1} p = \lambda \sum_ {k = 0} ^ {\infty} (I - \lambda A) ^ {k} p \approx \lambda \sum_ {k = 0} ^ {K} (I - \lambda A) ^ {k} p m \tag {30}
$$
where  $\lambda$  is a hyperparameter that ensures convergence and  $K$  is a hyperparameter for precision control.
AID-CG [12] approximates the hypergradient by solving the system of homogeneous linear equations  $Ax = p$  with conjugate gradient.
J Details regarding implementation, hyperparameters, query points, and computational times
J.1 Implementation and hyperparameters
For each PDE, we use a multilayer perceptron (MLP) for the PINN model and another MLP of the same architecture for EffiPICProp. Query points are collected in a grid manner. For each query point, PICProp starts with standard PINN training with a randomly initialized virtual boundary condition. We refer to this as the warmup phase. Subsequently, PICProp alternates between performing a number of inner steps to update the PINN model and computing the hyper-gradient for a meta step to update the virtual boundary condition. The inner/outer learning rate is selected from  $\{0.001, 0.01, 0.1\}$ . The inner/outer optimizer is selected from SGD and Adam. The hyper-gradient method is selected from Neumann Series (NS), Conjugate Gradient (CG), Reverse; see more details in Appendix I. The numbers of warmup/inner/meta steps are tuned to balance the stability of bi-level training and the computational cost. For the pedagogical example, as six points are too few for train-validation splitting, we manually set  $\lambda = 1$  for EffiPICProp. In the rest of the examples, we split  $10\%$  of the training data as the validation set to select the best  $\lambda$  from  $\{0.0, 0.25, 0.5, 0.75, 1.0\}$ . The hyperparameters are summarized in Table J.1.
Table J.1: Summary of implementation details, and utilized hyperparameters and architectures in the experiments.
Pedagogical	Poisson	Burgers
MLP arch.	32 × 2	20 × 8	20 × 8
# BC points	2	40	256
# query points	6	11 × 11	41 × 6
inner optim.	Adam	Adam	Adam
inner lr	0.001	0.001	0.001
meta optim.	SGD	SGD	SGD
meta lr	0.01	0.001	0.001
hypergrad	NS	Reverse	Reverse
# warmup steps	2,000	4,000	20,000
# inner steps	500	20	100
# meta steps	50	200	500
λ	1.0	1.0	0.0
J.2 Query Points
The query points are summarized in Figure J.1.
Figure J.1: Visualization of the query points on top of the exact solutions of Equations 11 and 13. (a) 2-D Poisson equation: 121 equally spaced points on a grid. (b) Burgers equation: 246 equally spaced points on a grid.
J.3 Computational times
In our computational experiments, all methods were run serially and thus, the runtimes shown below are much larger than the required ones for using these methods in practice. For this reason, we have also included a "pseudo-parallel" runtime that shows the runtimes if parallel processing had been used instead. The runtimes for the examples of Section 5 are summarized in Table J.2.
Table J.2: Computational time for different experiments.
Experiments	ES	PICProp	SimPICProp	EffiPICProp
Pedagogical example (CPU)	42 hrs for 5,000 trials	1.7 hrs for 41 queries	16 mins (40s + 6 PICProp queries)	18 mins (40s per lambda + 6 PICProp queries)
2D Poisson equation (GPU)	14 hrs for 1,000 trials	400s per query	-	14 hrs (180s per lambda + 121 PICProp queries), 10 mins for pseudo-parallel
70 hrs for 5,000 trials
1D time-dependent Burgers equation (GPU)	347 hrs for 2,500 trials	5,000s per query	-	14 hrs (2000s per lambda + 246 PICProp queries), 2 hrs for pseudo-parallel
694 hrs for 5,000 trials
K.1 Pedagogical example
K.1.1 Learning Curves
Figure K.1 provides the meta-learning curves corresponding to the CI predictions of Figure 2 (main text). Figure K.2 provides the results of all considered meta-learning methods and of ES on six query points. We conclude that the combination of Neumann Series (NS) and SGD stably converges to the reference result of ES.
K.1.2  $\lambda \mathbf{s}$
Figure K.3 provides comparison among PICProp and EffiPICProp with various  $\lambda$ s. Notice that EffiPICProp with  $\lambda = 0$  is referred to as SimPICProp in Figure 3. The mean squared errors (MSE) between PICProp CI and EffiPICProp CI are summarized in Appendix K.1.4. We conclude that EffiPICProp CI with  $\lambda = 1.0$  is the best approximation of PICProp CI.
K.1.3 SimPICProp v.s. EffiPICProp
Figure K.4 and Figure K.5 provide the comparison between SimPICProp and EffiPICProp on  $T^2$  and Hoeffding boundary CI, respectively.
K.1.4 Validity of EffiPICProp CIs
Although it has been already presented in Figure 2 that PICProp and EffiPICProp predictions well match the ES results, we conduct the following experiment to further verify the validity of EffiPICProp CI predictions.
We independently sampled 500 boundary conditions and propagate the corresponding  $\chi^2$  CIs using EffiPICProp to construct CIs for the entire domain. According to the definition of joint CIs (see Definition 3.1), an instance of CI is valid if it contains the solution  $u(x)$  for any  $x\in [-1,1]$ . In this regard, we uniformly sampled 101 data points from  $[-1,1]$  and regard an instance of CI as valid if the solution was enclosed within the interval for these data points. Further, we run vanilla PINNs with clean boundary data and use the max absolute error among these 101 data points as  $\eta$ . As a result, 486 out of these 500  $(97.2\% >95\%)$  instances of CI are valid, which empirically indicates that EffiPICProp CI predictons are valid and conservative.
Figure K.1: Pedagogical example: The learning curve of PICProp with Neumann Series (NS) and  $SGD$  on six query points. Comparisons with exhaustive search results for solving the bi-level optimization problem for each query point are also provided.
Figure K.2: Pedagogical example: Various combinations of hyper-gradient methods with meta-optimizers and comparisons with exhaustive search results for solving the bi-level optimization problem for each query point.
λ	0.0	0.25	0.5	0.75	1.0
MSE	1.4E-3	2.2E-4	2.2E-4	1.7E-4	1.8E-5
Table K.1: Mean Squared Errors between PICProp CI and EffiPICProp CI. EffiPICProp CI with  $\lambda = 1.0$  is the best approximation of PICProp CI.
K.2 2-D Poisson equation
Figure K.6 includes slices along the  $y$  axis and accompanies Figure 4 (main text), which includes slices along the  $x$  axis. Heatmaps with CIs over the entire domain are presented in Figure K.7.
Figure K.3: Pedagogical example,  $\chi^2$ : Comparison among PICProp and EffiPICProp with various  $\lambda$ s. Notice that EffiPICProp with  $\lambda = 0$  is referred to as SimPICProp in Figure 3
Figure K.4: Pedagogical example,  $T^2$ : Although the results of both SimPICProp and EffiPICProp match PICProp results on the query points, EffiPICProp generalizes better on unseen locations.
K.3 Burgers equation
Figures K.8 & K.9 provide the meta-learning curves corresponding to the CI predictions of Figure 5 (main text).
L Comparison with Bayesian PINNs