Title: Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks
Venue: NeurIPS 2023
Accepted: yes
Keywords: numerical methods

deep networks

evolution equations

scientific computing

partial differential equations

model reduction
Abstract: Training neural networks sequentially in time to approximate solution fields of time-dependent partial differential equations can be beneficial for preserving causality and other physics properties; however, the sequential-in-time training is numerically challenging because training errors quickly accumulate and amplify over time. This work introduces Neural Galerkin schemes that update randomized sparse subsets of network parameters at each time step. The randomization avoids overfitting locally in time and so helps prevent the error from accumulating quickly over the sequential-in-time training, which is motivated by dropout that addresses a similar issue of overfitting due to neuron co-adaptation. The sparsity of the update reduces the computational costs of training without losing expressiveness because many of the network parameters are redundant locally at each time step. In numerical experiments with a wide range of evolution equations, the proposed scheme with randomized sparse updates is up to two orders of magnitude more accurate at a fixed computational budget and up to two orders of magnitude faster at a fixed accuracy than schemes with dense updates.

Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks

Abstract

Training neural networks sequentially in time to approximate solution fields of timedependent partial differential equations can be beneficial for preserving causality and other physics properties; however, the sequential-in-time training is numerically challenging because training errors quickly accumulate and amplify over time. This work introduces Neural Galerkin schemes that update randomized sparse subsets of network parameters at each time step. The randomization avoids overfitting locally in time and so helps prevent the error from accumulating quickly over the sequential-in-time training, which is motivated by dropout that addresses a similar issue of overfitting due to neuron co-adaptation. The sparsity of the update reduces the computational costs of training without losing expressiveness because many of the network parameters are redundant locally at each time step. In numerical experiments with a wide range of evolution equations, the proposed scheme with randomized sparse updates is up to two orders of magnitude more accurate at a fixed computational budget and up to two orders of magnitude faster at a fixed accuracy than schemes with dense updates.

1 Introduction

In science and engineering, partial differential equations (PDEs) are frequently employed to model the behavior of systems of interest. For many PDEs that model complicated processes, an analytic solution remains elusive and so computational techniques are required to compute numerical solutions.

Global-in-time training There have been many developments in using nonlinear parameterizations based on neural networks for numerically approximating PDE solutions. These include techniques such as the Deep Galerkin Method [46], physics-informed neural networks (PINNs) [41], and others [4, 21, 53, 15]; as well as early works such as [11, 42]. In most of these methods, a neural network is used to represent the solution of a (time-dependent) PDE over the whole space-time domain. For this reason they are termed global-in-time methods in the following. To approximate the solution, the neural network is trained to minimize the PDE residual on collocation points sampled from the spacetime domain, which requires solving a large-scale optimization problem that can be computationally expensive. Additionally, the solutions learned by global-in-time methods can violate causality, which can become an issue for complex problems that rely on preserving physics [27]. We note that neural networks have been used for approximating PDE solutions in various other ways, such as learning specific component functions [23, 31, 43], finding closure models [2, 25, 50], de-noising [44], and for surrogate modeling [33, 32, 18]. However, we are interested in this work in using neural networks for directly approximating PDE solutions.

Sequential-in-time training with the Dirac-Frenkel variational principle In this work, we follow the Dirac-Frenkel variational principle, which has been used for numerical methods in the field of quantum dynamics for a long time [10, 17, 26, 34, 28] and for dynamic-low rank and related solvers [24, 45, 35, 39, 38, 22]. Instead of globally approximating a PDE solution in time, the Dirac-Frenkel variational principle allows a sequential-in-time training that adapts a nonlinear parameterization, such as a neural network, over time. In contrast to classical numerical methods in vector spaces, the approximate solution in Dirac-Frenkel schemes is allowed to depend nonlinearly on its parameters and so to lie on a smooth manifold. The update to the nonlinear parameterization is calculated at each time step according to the orthogonal projection of the dynamics onto the tangent space of the manifold induced by the nonlinear parameterization. The Dirac-Frenkel variational principle has been adapted for the nonlinear approximation of PDEs with neural networks. In particular [14, 1, 7] formulate a sequential-in-time method based on the Dirac-Frenkel variational principle.

The neural network represents the PDE solution at a point in time. The time-dependence then arises by allowing the parameters-the weights and biases of the network-to vary in time. The network parameters are then evolved forward according to the time dynamics which govern the PDE. This is in contrast to global-in-time methods, in which time enters the network as an additional input variable. By construction, an approximate solution obtained with a sequential-in-time method is causal, in that the solution at future times depends only on the solution at the current time.

Although these methods have demonstrated success in solving various PDEs [34, 14, 7, 54, 16, 8, 30], there are open challenges: First, the sequential-in-time training is prone to overfitting which can lead to a quick accumulation of the residual over time. Second, the local training step has to be repeated at each time step, which can be computationally costly, especially with direct solvers that have costs increase quadratically with the number of network parameters. The work [16] proposes to address the two issues by using iterative solvers, instead of direct ones, and by re-training the network occasionally over the sequential-in-time training. We show with numerical experiments below that the re-training of the network can be computationally expensive. Additionally, the performance of iterative solvers depends on the condition of the problem, which can be poor in the context of sequential-in-time training.

Our approach and contributions: Randomized sparse updates for schemes based on the DiracFrenkel variational principle We build on the previous work in sequential-in-time methods following a similar set up as [7] based on the Dirac-Frenkel variational principle. Where all previous methods solve local training problems that update every parameter of the network at each time step, we propose a modification such that only randomized sparse subsets of network parameters are updated at each time step:

(a) The randomization avoids overfitting locally in time and so helps preventing the error from accumulating quickly over the sequential-in-time training, which is motivated by dropout that addresses a similar issue of overfitting due to neuron co-adaptation.

(b) The sparsity of the updates reduces the computational costs of training without losing expressiveness because many of the network parameters are redundant locally at each time step.

Our numerical experiments indicate that the proposed scheme is up to two orders of magnitude more accurate at a fixed computational cost and up to two orders of magnitude faster at a fixed accuracy.

We release our code implementation here: https://github.com/julesberman/RSNG

2 Sequential-in-time training for solving PDEs

2.1 Evolution equations, Dirac-Frenkel variational principle, Neural Galerkin schemes

Given a spatial domain X ⊆ R d , and a time domain T = [0 , T ) ⊆ R , we consider a solution field u : T × X → R so that u ( t, · ) : X → R is in a function space U at each time t , with dynamics

where u 0 ∈ U is the initial condition and f can include partial derivatives of u to represent PDEs. We focus in this work on Dirichlet and periodic boundary conditions but the following approach

tangent space

M

Θ

f

(

θ

(

t

))

ˆ

u

(

·

;

θ

(

t

))

spanning set

sparse subset

can be applied with, e.g., Neumann boundary conditions as well [7]. One approach for imposing Dirichlet boundary conditions is by choosing parameterizations that satisfy the boundary conditions by definition [48].

Sequential-in-time training methods approximate u with a nonlinear parameterization such as a neural network ˆ u : X × Θ → R , where the parameter vector θ ( t ) ∈ Θ ⊆ R p depends on time t ; the parameter θ ( t ) has p components and enters nonlinear in the second argument of ˆ u . The residual of (1) at time t is where we applied the chain rule to ∂ t ˆ u ( · ; θ ( t )) to formally obtain ˙ θ ( t ) . Methods based on the Dirac-Frenkel variational principle [10, 17, 34] seek ˙ θ ( t ) such that the residual norm is minimized, which leads to the least-squares problem

in the L 2 ( X ) norm ∥ · ∥ L 2 ( X ) over X . The least-squares problem (3) gives a ˙ θ ( t ) such that the residual is orthogonal to the tangent space at ˆ u ( · ; θ ( t )) of the manifold M Θ = { ˆ u ( · ; θ ) | θ ∈ Θ } induced by the parameterization ˆ u ; see Figure 1. Schemes that solve (3) over time have also been termed Neural Galerkin schemes [7] because (3) can be derived via Galerkin projection as well.

The tangent space at ˆ u ( · ; θ ( t )) is spanned by the spanning set { ∂ θ i ˆ u ( · ; θ ( t )) } p i =1 , which are the component functions of the gradient ∇ θ ( t ) ˆ u ( · ; θ ( t )) ; it is important to stress that { ∂ θ i ˆ u ( · ; θ ( t )) } p i =1 is not necessarily a basis of the tangent space because it can contain linearly dependent functions and be non-minimal. The least-squares problem (3) can be realized by assembling a matrix whose columns are the gradient sampled at n ≫ p points x 1 , . . . , x n ∈ X resulting in J ( θ ( t )) = [ ∇ θ ( t ) ˆ u ( x 1 ; θ ( t )) , . . . , ∇ θ ( t ) ˆ u ( x n ; θ ( t ))] T ∈ R n × p , which is a batch Jacobian matrix to which we refer to as Jacobian for convenience in the following. Additionally, we form the right-hand side vector f ( θ ( t )) = [ f ( x 1 ; θ ( t )) , . . . , f ( x n ; θ ( t ))] T ∈ R n and thus the least-squares problem

The choice of the points x 1 , . . . , x n is critical so that solutions of (4) are good approximations of solutions of (3); however, the topic of selecting the points x 1 , . . . , x n goes beyond this work here and so we just note that methods for selecting the points exist [1, 7] and that we assume in the following that we select sufficient points with n ≫ p to ensure that solutions of (4) are good approximations of solutions of (3)

2.2 Problem Formulation

Challenge 1: Parameters are redundant locally in time. Typically, parameterizations ˆ u based on deep neural networks lead to Jacobian matrices J ( θ ( t )) that are low rank in the least-squares problem (4); see, e.g., [37] and Figure 2(a). In our case, a low-rank matrix J ( θ ( t )) means that components in ˙ θ ( t ) are redundant, because we assume that the samples x 1 , . . . , x n are sufficiently rich. Even if J ( θ ( t )) is low rank and thus the components in ˙ θ ( t ) are redundant, the problem (4) can still be

(a) low-rankness of Jacobian

(b) residual growth

(c) co-adaptation of neurons

solved with standard linear algebra methods such as the singular value decomposition (SVD) because they compress the matrix J ( θ ( t )) and regularize for, e.g., the minimal-norm solution; however, the costs of performing the SVD to solve (4) scales as O ( np 2 ) , and thus is quadratic in the number of parameters p . This means that a redundancy in ˙ θ ( t ) of a factor two leads to a 4 × increase in the computational costs. Note that the problem typically is poorly conditioned because J ( θ ( t )) is low rank, which makes the direct application of iterative solvers challenging.

Challenge 2: Overfitting leads to high residual over time. The residual from solving (4) can rapidly increase over time which in turn increases the overall error. This indicates that the tangent space along the trajectory θ ( t ) becomes ill suited for approximating the right-hand side vector f ( θ ( t )) in (4). We compare the residual of the least-squares problem (4) that is obtained along a trajectory of θ ( t ) from sequential-in-time training with the schemes above to the residual of (4) from a network that is fit to the true solution at each point in time; details in Appendix A.1. As shown in Figure 2(b), a lower residual is achieved by the network that is fit to the true solution.

We aim to understand this phenomenon through the lens of overfitting: the sequential-in-time training can be thought of as successive fine-tuning, in the sense that at each time step we must make a small update to our parameters to match the solution at the next time step. However, fine-tuning is well known to be prone to over-fitting and model degeneration [3]. In the setting considered in this work, overfitting means that the representation ˆ u ( · ; θ ( t )) does not generalize well to the next time step. Not generalizing well means that a local change to θ ( t ) is insufficient to move ˆ u ( · ; θ ( t )) according to the desired update given by ˙ θ ( t ) to match the right-hand side f ( θ ( t )) , which implies that a large residual is incurred when solving (4). A common approach to prevent overfitting is dropout [47], especially when applied to fine-tuning tasks with dropout variants proposed in [3, 29], while other approaches are formulated specifically around sparse updates [49, 52]. Dropout is motivated by the observation that dense updates to parameters in neural networks can cause overfitting by leading neurons to co-adapt. Typically, co-adaptation is characterized by layer-wise outputs with high covariance [9]. In the case of sequential-in-time training with the schemes discussed above, co-adaptation implies the columns of the Jacobian matrix J ( θ ( t )) are correlated and thus close to linearly dependent. So as neurons co-adapt, component functions of the gradient become redundant and may be less suited for approximating f ( θ ( t )) causing the high residual for the least-squares problem; see Figure 2(b). This could also be characterized by the ill conditioning issue pointed out in [16]. We see empirical evidence of co-adaptation in Figure 2(c), where we plot component functions of the gradient and see that they are strongly correlated at the end time T .

3 Randomized Sparse Neural Galerkin (RSNG) schemes

We introduce randomized sparse Neural Galerkin (RSNG) schemes that build on the Dirac-Frenkel variational principle to evolve network parameters θ ( t ) sequentially over time t but update only sparse subsets of the components of θ ( t ) and randomize which components of θ ( t ) are updated. The sparse updates reduce the computational costs of solving the least-squares problem (4) while taking advantage of the low rank structure of J ( θ ) which indicates components of the time derivative ˙ θ ( t ) are redundant and can be ignored for updating θ ( t ) without losing expressiveness. The randomization of which components of θ ( t ) are updated prevents the overfitting described above.

3.1 Randomized sketch of residual

To define the sketch matrix S t , let e 1 , . . . , e p be the p -dimensional canonical unit vectors so that e i has entry one at component i and zeros at all other components. We then define s independent and identically distributed random variables ξ 1 ( t ) , . . . , ξ s ( t ) that depend on time t . The distribution of ξ i ( t ) is π , which is supported over the set of indices { 1 , . . . , p } . The random matrix S t of size p × s is then defined as S t = [ e ξ 1 ( t ) , . . . , e ξ s ( t ) ] . The corresponding sketched residual analogous to (2) is

where now ˙ θ s ( t ) ∈ R s is of dimension s ≪ p .

3.2 Projections onto randomized approximations of tangent spaces

Using the sketch matrix S t , we obtain from the spanning set { ∂ θ i ˆ u ( · ; θ ( t )) } p i =1 of component functions of ∇ θ ˆ u ( · ; θ ( t )) a subset { ∂ θ ξ i ( t ) ˆ u ( · ; θ ( t ) } s i =1 with s functions. The set { ∂ θ ξ i ( t ) ˆ u ( · ; θ ( t ) } s i =1 spans at least approximately the tangent space at ˆ u ( · ; θ ( t )) of M Θ but has only s ≪ p elements. The motivation is that the full spanning set { ∂ θ i ˆ u ( · ; θ ( t )) } p i =1 contains many functions that are close to linearly dependent (Jacobian is low rank) and thus sub-sampling the component functions still gives reasonable tangent space approximations that preserves much of the expressiveness; see Figure 1. While the low rankness depends on the complexity of the problem and parametrization, we observe low rankness in all our examples; see Appendix A.1 for further discussion.

We now introduce a least-squares problem based on the sparse spanning set { ∂ θ ξ i ( t ) ˆ u ( · ; θ ( t ) } s i =1 that is analogous to the least-squares problem problem based on the full spanning set given in (4). We seek ˙ θ s ( t ) ∈ R s with s components that solves

To obtain ˙ θ ( t ) to update θ ( t ) , we set ˙ θ ( t ) = S t ˙ θ s ( t ) . Thus, the components of ˙ θ ( t ) that are selected by S t are set to the corresponding value of the component of ˙ θ s ( t ) and all other components are set to zero, which means that the corresponding components of θ ( t ) are not updated. We can realize (6) the same way as the full least-squares problem in (4) by using the full Jacobian matrix and S t to define the sparse Jacobian matrix as J s ( θ ( t )) = J ( θ ( t )) S t and the right-hand side vector f ( θ ( t )) analogous to Section 2 to obtain the discrete least-squares problem

The choice of the distribution π is critical and depends on properties of the Jacobian matrix J ( θ ( t )) . Distributions based on leverage scores provide tight bounds with regard to the number of columns one needs to sample in order for the submatrix to be close to an optimal low rank approximation of the full matrix with high probability [13]. But these distributions can be expensive to sample from. Instead, uniform sampling provides a fast alternative.

The number of columns will not grow too quickly if the full matrix is sufficiently incoherent [19]. This means some columns do not carry a disproportionate amount of information relative to other columns. We numerically see that in our case the Jacobian matrix J ( θ ( t )) is sufficiently incoherent. Thus we can choose a uniform distribution over the set of indices { 1 , . . . , p } to get the benefits of low rank approximation in a computationally efficient way.

3.3 Discretizing in time

We discretize the time interval T with K ∈ N regularly spaced time steps 0 = t 0 < t 1 < · · · < t K = T with δt = t k -t k -1 for k = 1 , . . . , K . At time t 0 , we obtain θ (0) ∈ R p by fitting the initial condition u 0 . We then update

for k = 1 , . . . , K so that θ ( k ) is the time-discrete approximation of θ ( t k ) and thus ˆ u ( · ; θ ( k ) ) approximates the solution u at time t k . The sparse update ∆ θ ( k -1) s approximates ˙ θ s ( t k -1 ) and is obtain by the time-discrete counterpart of (7), which is given by

if time is discretized with the forward Euler method. Other discretization schemes can be used as well, which then lead to technically more involved problems (9) that remain conceptually similar though. The sparse update is lifted to ∆ θ ( k -1) = S k ∆ θ ( k -1) s so that the update (8) can be computed.

3.4 Computational procedure of RSNG

We describe the proposed RSNG procedure in algorithmic form in Algorithm 1. We iterate over the time steps k = 1 , . . . , K . At each time step, we first sketch the Jacobian matrix by creating a submatrix from randomly sampled columns. Notice that S k need not actually be assembled as its action on the Jacobian matrix can be accomplished by indexing. We then solve the least-squares problem given in (9) using our sketched Jacobian to obtain ∆ θ ( k -1) s . A direct solve of this system dominates the computational cost of making a time step and scales in O ( ns 2 ) time. The components of ∆ θ ( k -1) corresponding to the indices that have not been selected are filled with zeros. We then update the parameter θ ( k -1) to θ ( k ) via ∆ θ ( k -1) .

The whole integration process scales as O ( T δt ns 2 ) in time.

4 Numerical experiments

We demonstrate RSNG on a wide range of evolution equations, where speedups of up to two orders of magnitude are achieved compared to comparable schemes with dense updates. We also compare to global-in-time methods, where we achieve up to two orders of magnitude higher accuracy.

4.1 Setup and equations

Examples We now describe the details of the PDEs that we use to evaluate our method. We choose these particular setups to test our method on a diverse set of challenges including problems with global and local dynamics and solutions with sharp gradients and fine grained details. For visualization of the solutions of these equations see the Appendix A.4.

Reaction-diffusion problem modeled by Allen-Cahn (AC) equation: The Allen-Cahn equation models prototypical reaction diffusion phenomena and is given as,

We choose ϵ = 5 × 10 -3 , with periodic boundary condition X = [0 , 2 π ) and initial condition,

This initial condition results in challenging dynamics that are global over the spatial domain.

Flows with sharp gradients described by Burgers' equation: The Burgers' equation is given by,

We choose ϵ = 1 × 10 -3 , with periodic boundary condition X = [ -1 , 1) and initial condition,

The corresponding solution field has sharp gradients that move in the spatial domain over time, which can be challenging to approximate.

Charged particles in electric field: The Vlasov equation describes the time evolution of collisionless charged particles under the influence of an electric field. The equation models the distribution of such particles in terms of their position and velocity. We consider the case of one position dimension and one velocity dimension, making our domain X ⊆ R 2 . The equation is given by,

where x is the position, v is the velocity and ϕ is the electric field. We consider the case with periodic boundary condition X = [0 , 2 π ) × [ -6 , 6) and initial condition,

with a fixed electric field ϕ ( x ) = cos( x ) . This particular setup evolves into a distribution with fine grained details along a separatrix surrounding the potential well.

Setup We parameterize with a feed-forward multi-layer perceptron. All our networks use linear layers of width 25 followed by non-linear activation functions, except the last layer which has no activation and is of width 1 so that ˆ u ( x , θ ( t )) ∈ R . To vary the number of total parameters p , we vary the depth of networks ranging from 3-7 layers. We use rational activation functions which in our experiments allowed for fitting initial conditions faster and more accurately than a standard choice such as tanh or ReLU [5]. To enforce periodic boundary conditions, we modify the first layer so that it outputs periodic embeddings as in [7]; for details see Appendix A.5. The periodic embedding ensures that the boundary conditions are enforced exactly. For additional details on enforcing other types of boundary conditions (e.g. Dirichlet and Neumann) exactly in neural networks see [12, 7, 48]. We sample points from the domain on an equidistant grid. For time integration we use a RK4 scheme with a fixed time step size. The time step sizes are 5e -3 , 1e -3 , 5e -3 and we integrate up to end time 4 , 4 , and 3 for the Allen-Cahn, Burgers', and Vlasov equations, respectively. All error bars show + / -two standard errors over three random realizations which results in different sketching matrices at each time step. Relative errors are computed over the full space-time domain, unless the plot is explicitly over time.

All gradients and spatial derivatives are computed with automatic differentiation implemented in JAX [6]. All computations are done in single precision arithmetic which is the default in JAX. All runtime statistics were computed on the same hardware, a Nvidia Tesla V100 w/ 32 GB memory. All additional hyperparameters are described in Appendix A.5.

4.2 Results

RSNG achieves higher accuracy than schemes with dense updates at same computational costs In Figure 3 we plot the relative error over time. The curves corresponding to 'dense updates' use a 3 layer network and integration is performed using dense updates. For RSNG, we use a 7 layer network and integrate with sparse updates, setting the number of parameters we update, s , equal to the total number of parameters in the 3 layer network and thus equal to the number of parameters that are updated by 'dense updates.' Thus, the comparison is at a fixed computational cost. The error achieved with RSNG is one to two orders of magnitude below the error obtained with dense updates, across all examples that we consider. In Figure 4(a), we see that as we increase the network size,

(b) Burgers'

(a) fix sparsity s , vary #network parameters p

(b) fix #network parameters

p

, vary sparsity

s

Figure 4: (a) RSNG benefits from the additional expressiveness of larger networks (larger p ) while only using a fixed number of parameters (fixed s ) at each time step. (b) As we decrease the number of parameters s in the sparse update, but keep the total number of parameters p of the network the same, we achieve lower errors than dense updates. Thus, RSNG outperforms dense updates while incurring lower computational costs. Error bars generated over random sketch matrices, S t .

the relative error decreases as the sparse updates allow us to exploit the greater expressiveness of larger networks while incurring no additional computational cost in computing (9). But we note that increasing the size of the full network will make computations of J ( θ ) and f ( θ ) more expensive because of higher costs of computing gradients. However, for the network sizes that we consider in this work, this effect is negligible compared to the cost of solving (9).

RSNG achieves speedups of up to two orders of magnitude In Figure 5(a), we compare the runtime of RSNG to the runtime of a scheme with dense updates that uses a direct solver and to the runtime of a scheme with dense updates that uses an iterative solver as proposed in [16]. The time is computed for Burgers' equation and the sparsity s of RSNG is chosen such that all methods reach a comparable level of error. We find that RSNG is faster than direct solves with dense updates by two orders of magnitude and faster than the iterative solver by one order of magnitude.

The results show that while using an iterative solver as in [16] does speed up the method relative to direct solves with dense updates, it can still be quite slow for networks with many parameters p . Additionally, convergence of the iterative method given in [16] requires a number of hyperparameters to be chosen correctly, which may require an expensive search or a priori knowledge about the solution. Note that our RSNG method does not preclude the use of an iterative method to speed up the least-squares solves further.

(b) error versus runtime

(a) runtime at 2e -4 relative error with s = 125

To address the overfitting problem, the work [16] refits the network to the current approximate solution from a random initialization periodically during time integration. In Figure 5(b), we show the relative error versus the runtime for the iterative solver with various numbers of refits and for RSNG at different sparsity s . While refitting the network can reduce the relative error, it incurs a high computational cost. By contrast, for appropriate sparsity s , RSNG outperforms the method given in [16] in both speed and accuracy.

Varying sparsity s at fixed number of total parameters p in network We now study the effect of varying the sparsity s (i.e., number of parameters updated by sparse updates) for a fixed network of total size p , in this case a 7 layer network. In Figure 4(b), we see that for a network of fixed size, sparse updates can reduce the relative error by about 2 -3 × when compared to dense updates. This is notable as the computational costs decrease quadratically with s . Thus, the combination of sparsity and randomized updates in RSNG can deliver both improved performance and lower computational cost. We see that at the beginning, when the number s is too small, the expressiveness suffers and the error becomes large. This is because if s is less than the rank of the dense Jacobian then the sparsified Jacobian will necessarily have less representational power. However, we stress that RSNG is robust with respect to s in the sense that for a wide range of s values the error is lower than for dense updates.

The high error when performing dense updates s = p in Figure 4(b) for Allen-Cahn and Burgers' equation is due to the overfitting problem described in Section 2.2. As updates become denser, the method is more likely to overfit to regions of the parameter space in which the Jacobian, J ( θ ) , is ill suited for approximating the right-hand side f at future time steps (see Section 2). We can see this explicitly in Figure 6 where we plot the residual over time for sparse and dense updates on the Allen-Cahn equation. Initially, the dense updates lead to a lower residual. This makes sense as they begin at the same region of parameters space. But as the two methods navigate to different regions of parameters space, we see RSNG begins to incur a lower residual relative to dense updates. This indicates that RSNG ameliorates the problem of overfitting and so leads to a lower residual as shown in Figure 6(b).

Comparison with global-in-time methods We compare our method to global-in-time methods which aim to globally minimize the PDE residual over the entire space-time domain. We compare to the original PINN formulation given in [41]. Additionally we compare to a variant termed Causal PINNs, which impose a weak form of time dependence through the loss function [51]. We select this variant as it claims to have state of the art performance among PINNs on problems such as the Allen-Cahn equation. In Table 1, we see that our sequential-in-time RSNG method achieves a higher accuracy by at least one order of magnitude compared to PINNs. Additionally, in terms of computational costs, RSNG outperforms both PINN variants, as their global-in-time training is expensive and requires many residual evaluations. We note that the training time of PINNs is directly

(a) residual at early times

(b) residual over full time

-

dependent on the number of optimization iterations and thus they can be trained faster if one is willing to tolerate even higher relative errors.

5 Conclusions, limitations, and future work

In this work, we introduced RSNG that updates randomized sparse subsets of network parameters in sequential-in-time training with the Dirac-Frenkel variational principle to reduce computational costs while maintaining expressiveness. The randomized sparse updates are motivated by a redundancy of the parameters and by the problem of overfitting. The randomized sparse updates have a low barrier of implementation in existing sequential-in-time solvers. The proposed RSNG achieves speedups of up to two orders of magnitude compared to dense updates for computing an approximate PDE solution with the same accuracy.

Current limitations leave several avenues for future research: first, as discussed in 3.2, uniform sampling is only appropriate when the Jacobian matrix is of low coherence. Future work may investigate more sophisticated sampling methods such as leverage score and pivoting elements of rank revealing QR. Second, there are problems for which overfitting with dense updates is less of an issue; e.g., the charged particles example in our work. Note that due to the sparsity of the updates, RSNG still achieves a speedup compared to dense updates for the same accuracy for this example though. However, more work is needed to better understand and mathematically characterize which properties of the problems influence the overfitting issue.

We make a general comment about using neural networks for numerically solving PDEs: The equations discussed in this paper are standard benchmark examples used in the machine-learning literature; however, for these equations, carefully designed classical methods can succeed and often have lower runtimes than methods based on nonlinear parameterizations [20, 36]. While these equations provide an important testing ground to demonstrate methodological improvements, future work will extend these results to domains where classical linear methods struggle, e.g., high-dimensional problems and problems with slowly decaying Kolmogorov n -widths [15, 7, 40].

We do not expect that this work has negative societal impacts.

Acknowledgements The authors were partially supported by the National Science Foundation under Grant No. 2046521 and the Office of Naval Research under award N00014-22-1-2728. This work was also supported in part through the NYU IT High Performance Computing resources, services, and staff expertise.

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A Appendix

A.1 Details on Figure 2

For Figure 2a-b we look at quantities generated by fitting a network to the true solution at a point in time t . This is done in the same way we fit initial conditions described in A.5, but in this context the target function is taken to be the true solution at time t . For Figure 2(a), we compute the Jacobian of the network fitted to the true solution at each point in time and then plot its spectrum. For Figure 2(b), we take the network fitted to the true solution and compute the residual from the least-squares problem 4 to give the data points in the "Direct fit" line.

In Figure 7 we provide versions of Figure 2(a) from the main text but for all the equations considered. Vlasov is a more complex problem that exhibits a less sharp decay but is still distinctly rank deficient.

A.2 Details on Speed-Up

Here we provide additional results on the speed-up provided by RSNG for different equations. In Figure 8 we provide versions of Figure 5(a) from the main text but for all the equations considered. In Figure 9 we show how the runtime of Neural Galerkin schemes scales with s , thus showing the quadratic speed-up provided by RSNG as we reduce s .

A.3 Applications to High Dimensional Problems

Neural Galerkin schemes have been shown to be a useful approach to high dimensional PDEs [7]. In Figure 10 we demonstrate that RSNG may be applicable in these settings we well. We fit a neural network to a numerical solution of the Fokker-Planck equation in 5 dimensions; see [7, Section 5.4.1] for a description of the setup. The results show that for a sufficiently large network, the Jacobian has a low-rank structure as in the examples in the paper and a steep decay in the singular values so that random sketching strategies will likely be successful.

A.4 Ground Truth

The ground truth for the Allen-Cahn and Burgers' equations were generated using a spectral method with a fourth order integrator implemented in the spin solver as part of the Chebfun package in Matlab. We used a spatial grid of 10 000 points and time step size of 1e -3 .

The ground truth for the Vlasov equation was generated with a 4th order finite difference scheme in space and an RK4 time integration scheme. We sample 10 6 points over the full 2D space domain and a time step size of 1e -3 was used for time integration.

In Figure 11,12,13 we show plots of the ground truth solutions at the beginning, middle, and end of integration time for the equations we examine. We can see that they display the characteristics described in Section 4.1.

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A.5 Architecture and Hyperparameters

For an input x ∈ R d a periodic embedding layer with period P is defined with the optimization parameters a, ϕ, b ∈ R d as,

This operation is the repeated w times for different parameters a, ϕ, b where w denotes the width of the layer resulting in a output vector y ∈ R w

To fit the initial condition, we minimize the L 2 distance between the network and the initial condition as well as the L 2 distance between the first derivative of the network and the first derivative of the initial condition with respect to the spatial domain X . We evaluate the loss function over 10 000 equispaced points for Allen-Cahn and Burgers' equation and 200 000 points for Vlasov. We fit our initial conditions with two nonlinear solvers. First we run L-BFGS with 1000 iterations, then we use an Adam optimizer with the following hyperparameters,

iterations : 100 000

learning rate : 1e - 3

scheduler : cosine decay

decay steps : 500

The number of iterations and the number of points we sample are chosen to fit the initial condition to high accuracy to avoid polluting the results in our analysis with errors of fitting the initial condition.

To assemble the Jacobian matrix J ( θ ( t )) , the gradient of ˆ u is evaluated on samples generated from the spatial domain. If not noted otherwise, we use 10 000 equidistant points for Allen-Cahn and Burgers' and 200 000 equidistant points for Vlasov. For the time-optimized RSNG results in Table 1 (rows 4 and 8) we use 1000 equidistant points for Allen-Cahn and Burgers'. In the dense and sparse least-squares system we regularize the direct solver so as to avoid numerical instability. For this we set the rcond parameters in numpy implementation of lstsq . The values used are 1e -4 , 1e -4 , 1e -5 for Allen-Cahn, Burgers', and Vlasov respectively.

A.6 Global Methods Benchmark

Here we detail the training setups for our benchmarks of the global methods given in Table 1.

For all PINN experiments we sampled data on a grid with 100 points in the time domain and 256 points in the spatial domain. All PINNs were trained with the following hyperparameters:

optimizer : Adam then L-BFGS

spatial samples : 256

time samples : 100

activation : tanh

For the plain PINN experiments our architecture is a MLP with layers sizes as follows: [2, 128, 128, 128, 128, 1]. Boundary and initial conditions were enforced through a penalty in the loss function.

For the Causal PINNs we use the architecture described in the original paper, with periodic embedding and 'modified MLP layers.' The layer sizes were as follows: [ periodic embedding , 128, 128, 128, 128, 1]. Our implementation uses much of the original code provided in [51]. We used only one time-window for training as this is what was chosen in [51] for the Allen-Cahn equation. The tolerance hyperparameter, which controls the degree to which the loss function enforces causal training, was set to 100 and 50 for the Allen-Cahn and Burgers' equations respectively. Additionally the λ ic parameter, which controls the loss functions' weighting on the initial condition, was set to 100 and 1 for the Allen-Cahn and Burgers' equations respectively. These were both chosen via a hyperparameter search. For details on the context and meaning of these hyperparameters see the original paper [51].

The PINNs trained for Allen-Cahn used 1000 steps of Adam followed by 30000 steps of L-BFGS. For Burgers' equations we used 1000 steps of Adam followed by 60000 steps of L-BFGS. More steps are needed for Burgers' equation in order to sufficiently resolve the sharp gradient in the solution due to the low viscosity number. We similarly choose a smaller timestep for Burgers' in the Neural Galerkin schemes. We note that fewer optimization iterations can be used for training PINNs but this resulted in much larger errors in our experiments. In any case, widely varying the number optimization iterations and other PINNs hyperparameters did not achieve errors in the range of what RSNG achieves in these examples.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a so-called randomized sparse neural galerkin (RSNG) algorithm to solve time-dependent PDEs. The basic idea is to\nconsider parameterized functions as ansatz spaces, with parameters that vary in time. Consequently, the method is of the sequential-in-time learning algorithms rather than global-in-time learning algorithms such as plain vanilla PINNs. Computing PDE residuals with this class of ansatz functions and collocating the loss (at every time instant) naturally leads to a least-squares type linear algebra problem which can be solved either directly or indirectly and has been previously considered. The author identify problems with this approach and propose a novel remedy by considering randomized sparse instantiations of the projection, loosely along lines of dropout, but in a more L2 sense. The algorithm is illustrated with some benchmark examples for PDEs.\n\nSTRENGTHS:\n1. The premise of the paper is interesting. It is well-known that global-in-time methods such as PINNs are ill-conditioned and hard to train and the paper investigates a method of potentially avoiding some of the pitfalls of global-in-time methods. \n\n2. Randomized sparse algorithms can be efficient in the context described by the authors as long as certain conditions, such as incoherence that they loosely identify, are fulfilled. \n\n3. Numerical experiments clearly demonstrate that RNSG outperforms the dense least-squares method as well as baselines such as plain vanilla PINNs considerably.\n\nWEAKNESSES:\n1. Clarity: The paper could do with a rewrite where authors elaborate on the rather succinct notation, for instance in the appendix. In particular, to aid the reader, they could write out the neural networks in line 81, for instance for a simple one-hidden layer shallow neural network, as to how the time variable and the space variable exactly enter into Eqn (2) -- the reader will clearly see the distinct roles played by time here. Similarly, further rationale could be provided for the choice n >> p in deriving Eqn (4), one could for instance refer to it a quadrature for Eqn. (3). Caption of Figure 2 could be better explained, in particular what is refit in Figure 2(b) and what does co-adaptation mean and how does Figure 2 (c) shows this property. Quite a lot of heuristics is provided in a hand wavy manner that is potentially inscrutable to a reader -- for instance, what does the lines 173-176 mean ? \n\n2. Theory or rather the lack of it: The author do not attempt to provide any theoretical justification of their method. Some phrases are included here and there (e.g. line 162, lines 173-179 etc) but the reader feels that the authors know more about the theory then they present here. A discussion on these aspects could be illuminating. In particular, when is the method supposed to work and when not ? The lack of presentation of a theoretical perspective forces this reviewer to judge and criticize the paper on its empirical content as I proceed to do below. \n\n3. Empirical evaluation: The authors consider three time-dependent PDEs i.e., 1-d Allen-Cahn, 1-d viscous Burgers with a relatively low viscosity and a 2-d Vlasov-Poisson with a given potential and on a Cartesian domain. As such the problems are fairly well-studied benchmarks and the authors clearly show that their RNSG algorithm handily outperforms its dense foundation and an iterative version. However, the following limitations of the experiments can be identified: \n\n      a. The authors claim that it takes 18 mins for a PINN to obtain 6% errors for Allen-Cahn and almost 2 hours to obtain below 1% error for the simple viscous-Burgers example. This is unacceptably large and could very well be due to the use of ADAM as the optimizer. It is standard in PINNs literature to either use L-BFGS or start with ADAM and then switch to L-BFGS after say Order of 1000 iterations or so. This reviewer is curious if doing so reduces the run-time with PINNs ? In any case, the authors obtain significantly lower errors with RNSG -- so the timing is worth investigating. \n\n   b. Clearly, PINNs and even Casual PINNs (I am not convinced that it takes more than 6 and half hours to solve Burgers' equation with casual PINNs) are not the best conditioned to be the only baselines here. Rather I would suggest the following baselines: \n\n             b1) A method of lines version of PINNs (see the Raissi et al original PINNs paper) where the time is discretized with a RK4 solver and space with PINNs -- this is a fairer comparison as time is now iteratively updated with the PINNs and should be compared against your method, at least for some experiments. \n\n            b2) Alternatives to PINNs such as ODILs (Karnakov et al https://arxiv.org/abs/2205.04611) which directly collocate the PDE residual (with an implicit time-discretization) appear to be much better conditioned and could be a strong baseline for your method. \n\n         b3) The whole rationale would be out-compete standard numerical methods. Hence, it would be instructive to consider the computational cost of your ground truth generator. How much computational time does it take with this ground truth generator -- for all the experiments -- to reach comparable accuracy. Given that you require 13 mins to solve 1-D Allen-Cahn and 40 minutes to solve 1-D Burgers, traditional numerical methods would solve both these problems in a matter of secs, if not faster -- so where is the rationale for the use of your method ? One could argue that its rationale would be high-dimensional problems or complex geometries, which you don't consider empirically here. \n\n4. Scope: In its current version, it is unclear how this algorithm scales with spatial (or parametric) dimension -- is it fairly dimension independent ? If so, where is the evidence for that -- for instance, what is the computational time for the 2-D Vlasov-Poisson problem ? \n\nAnother issue pertaining to the scope is that the method relies on an explicit time-stepping integrator such as RK4 -- can it be used for stiff problems, with implicit time stepping ? Otherwise, you will require too many iterations in time, something that PINNs and other global-in-time methods don't have to. It would be useful to see if your algorithm can work with large time steps for such stiff problems. \n\n5. Comparison: The authors say that other works such as [1,7,14] have considered this approach and yet do not provide a clear comparison to these works. It is essential to distinguish between them and the current paper, in a related work section.\n\nQUESTIONS:\nIn addition to the questions above in the \"weaknesses\" section -- the following minor ones should be addressed:\n\n1. What is refit in Figure 2(b) -- it is also used in other figures ?\n\n2. What is computational cost of RNSG for 2-D Vlasov-Poisson ? How does it compare to PINNs, especially when they are optimized with L-BFGS. \n\n3. How do the results, particularly the choice of the sparsity parameter $s$ change with the viscosity $\\nu$ for viscous-Burgers' ? How low can $\\nu$ be to still allow for accurate approximation ? \n\n4. Please mention the value of $s$ in each of the figures (for instance in the option or the Appendix) where you have not mentioned the value.\n\nLIMITATIONS:\nIt is a mostly a reiteration of the section on Weaknesses:\n\n-- No theoretical underpinning of the algorithm.\n\n-- Lack of a clear description of the limitations of the algorithm.\n\n--- How are the collocation points $x_i$ chosen ? The authors refer to a prior work but do not contextualize it here.\n\n--- More baselines need to be evaluated -- in particular, the runtime of traditional numerical methods has to be mentioned to provide the reader with an idea of what is the potential pay-off with their method, if any ? \n\nThe basic idea is interesting and I am happy to reconsider my score if the authors address my questions appropriately.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper suggests a simple but effective way to train sequential-in-time models by randomly sampling weights to update. The method avoids overfitting and realizes speedup because only a small part of parameters is updated. The experiments show that the proposed model offers the best performance in terms of speed and accuracy.\n\nSTRENGTHS:\n* Although the method is simple, the idea is based on the reasonable assumption that parameters are redundant locally in time. Due to its redundancy, the Jacobian matrix is prone to be singular or stiff and takes a huge computational budget for training with the existing sequential-in-time methods. The proposed method elegantly addresses these issues by randomly sampling weights to update.\n* The experiments clearly demonstrate that the proposed method works efficiently in terms of computation time and accuracy compared to the considered baseline methods.\n\nWEAKNESSES:\n* Figure 5 (b) and (c) suggest that sampling could lead to instability (see also Questions). The reviewer recommends that the authors show predicted solutions in figures to assess the smoothness of the prediction. Sometimes smoothness is key because some downstream tasks may perform additional differentiation on the prediction.\n\nMinor points:\n* In the first equation in Equation (1), the domain of t should be defined in the open set (see, e.g., Jürgen Jost \"Partial Differential Equations ThirdEdition\" (2012) Chapter 1) to make differentiation possible everywhere in the domain. That's why we need the initial condition.\n\nQUESTIONS:\n* Is the method faster than classical numerical solvers (e.g., finite difference method)? It could be important to compare with the classical methods.\n* Figure 5 (b) and (c) suggest that sampling could lead to instability. What is the reason, and any ideas to address this? The reviewer guesses that the updated weights are not smooth in time, i.e., the indices of the weight matrix are drastically changed at a time $t$ and the next time $t + \\Delta t$. Could it be interesting to introduce some \"momentum\" to make the change of indices moderate in time?\n\nLIMITATIONS:\n* The authors addressed limitations. Another limitation could be handling boundary conditions other than periodic (e.g., Dirichlet and Neumann).", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors suggest an alternative optimization scheme for Neural Galerkin methods - a family of methods in which parameters of a neural network are optimized so that the gradients of the solution obey a given PDE with the notable quality being that rather than solving over all of space-time, the parameters of the network evolve in time to match the system evolution equation - in which only a subset of the surrogate model's parameters are updated at each time step. The author's show that this approach seems to lead to improved accuracy at a fixed FLOP budget and improved speed to a fixed accuracy threshold.\n\nSTRENGTHS:\n- The paper is well written and interesting.\n- Jargon is generally defined shortly after it is used.\n- As someone who is not very familiar with Neural Galerkin methods, it seemed that most questions that immediately came to mind after reading segments were answered quickly. \n- Overall, the paper was clear, compelling, and all claims were well-supported by the provided evidence.\n\nWEAKNESSES:\nThe main weakness of the paper is that while the paper is clearly providing value to a current research topic, it could do a better job of explaining where Neural Galerkin methods are (or could be) practically useful. The PINN comparison is enough for me to be comfortable recommending approval, but given the method requires knowledge of $f$, it seems like true comparison would be numerical solvers which, as the authors mention in the conclusion, are still quite a bit ahead of the PINN comparisons on these problems. Without that, it's difficult to see whether the paper would be interesting to a wider audience.\n\nLine-item Issues:\n- Figure 2 is not very clear. a) makes sense, but (b) could use a better captioning so that it can be understood independently of the text and (c) is unclear even after reading the text. \n- Figures 3, 4 are similar in that the caption is currently insufficient to explain the figure.\n\nQUESTIONS:\n- Were the PINN comparisons also performed with rational activation functions?\n- How do these methods compare against numerical methods?\n- Given the conclusion implies the comparison is unfavorable, it would be very valuable to see examples from the set of problems where numerical methods are expected to do poorly.\n\nLIMITATIONS:\nLimitations are well covered and no ethical issues are foreseen.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors propose a modified Neural Galerkin scheme for the solution of time-dependent PDEs, aiming at reducing computational cost and improving accuracy by avoiding overfitting.\n\nThe target solution $u(x,t)$ is approximated with a nonlinear parameterisation $\\theta(t)$, so that $u(x,t)\\approx \\hat{u}(x;\\theta(t))$ (this can be interpreted as a nonlinear form of reduced-order modelling). In practice, $\\theta(t)$ represents the parameters of a NN responsible for outputting $\\hat{u}(x;\\theta(t))$ for a given $x$. At each time-step $t$, the goal is to train the network to output the solution at the next time-step. As is classical with Galerkin methods (and derivative thereof), this is achieved by following a variational principle, whereby the norm of the residual of the PDE is minimised. This can be recast as a least-square minimisation problem in the unknown $\\dot\\theta$: once solved, the gradient of $\\theta$ is then used to guide the update of the NN.\n\nThe novel contribution of the paper lies in applying sketching to simplify the solution of this least-square problem. This is rooted in the observations that:\n- Often, the target matrix in the system $J(\\theta)$ (a collation of the gradients of $u$ wrt $\\theta$, sampled at specific collocation points) is low-rank, implying redundancy in the components of $\\theta$\n- Updating the whole $\\theta$ tends to overfit, which in turn leads to accumulation of errors and poor quality of the solution as time evolves.\nThis justifies reducing the space of parameters to update, and has the boosted effect of reducing computational time due to the effective shrinking of the least-square problem.\n\nThe performance of the novel method is verified on three benchmark PDEs (two 1D and one 2D), and expressed in terms of runtime and norm of residual. The proposed method outperforms both its dense (non-sketched) counterpart, and the global-in-time method (Causual) PINN.\n\nSTRENGTHS:\n- The sketching simplification at the core of the method is well-justified, also with experimental results. Moreover, I particularly appreciate the fact that the intuition behind using sketching is rooted both in algebraic considerations (pertaining rank of J), but is also interpreted through the lenses of regularisation techniques in NN (dropout). Both views are valid, and contribute to supporting the hypothesis\n- The paper is well written, neatly structured, with the right level of rigour and a good introduction to the topic\n\nWEAKNESSES:\n- Possibly the breadth of the contribution: after all, it represents a relatively small modification to an already existing method, and the modification itself takes inspiration from already-established techniques. Still, I appreciate this work, and the contribution seems valid and useful\n- Some experiments might be expanded to further substantiate the claims (see questions below), but this should be straightforward to address\n\nQUESTIONS:\n- How does Fig2a look like for Burgers’ and Vlasov? And if we vary the network size? I’m trying to understand whether the low-rankness observation is affected somehow by the type of problem/model considered, or if it is robust to these variables (Fig5a partially hints at this, but it would draw a fuller picture to include comparisons throughout the paper)\n- Line 257: can you report the actual value of the error in this case? Also, how does Fig4(a) look like for Allen-Cahn and Vlasov? This builds up on the previous question: I’m trying to understand how robust these findings are wrt problem chosen\n- Fig 4(b): how does time complexity scale with $s$? In my mind, this was supposed to be the bottleneck of the algorithm, and if you’re using a direct solver to solve the associated least-square system, complexity (and time) should scale quite heavily in $s$ (you mention \"quadratically\" a couple of times: what exactly are you using as solver?). But by eyeballing the datapoints in the plot, it looks almost linear? Am I missing something?\n- I liked Fig5(a): the fact that it’s U-shaped really fits the overfitting assumption (pun intended), and I think it really helps nailing down the validity of the method. I also like the fact that it shows that Vlasov kinda falls outside the trend - but this makes sense: it’s a more complex problem after all (if anything, because it’s 2D). Ultimately, I expect these graphs to be both problem- and NN- dependent: if you had chosen a NN with more parameters, likely at a certain point it would've started overfitting again, even for Vlasov. In any case, it’s extremely useful to know when a method might start breaking down, and it should be pointed out! \n\nMinor corrections\n- Fig3(d):\n    - For the x-label: I reckon $757$ is the fixed value of $s$? Just write $p (s=757)$ or something on those lines. Same for Fig5\n    - Please report an indication of scale (for example, by explicitly specifying a numerical value for the “Dense” tick)\n    - I reckon the relative error here refers to the end time $T=4$?\n    - Maybe they would look better if the figures shared the same scale for the y-axis? This way one could align them and compare better among them? Not sure if they’d get too squashed though\n- Line 237: I can see error bars only for Fig3(d). If for Fig(a-c) they’re too small to see, please state it explicitly. If you’re not reporting them, why not? Similarly for Fig5\n- Line 245: please clarify: “we plot the relative error [with respect to the underlying solution] over time”\n\nLIMITATIONS:\nThe main limitations I could identify (particularly, dependency on properties of the target Jacobian) are addressed and made explicit in the corresponding section", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

comments: SUMMARY:
The paper introduces Neural Galerkin scheme to sparsely update neural network over the sequential-in-time training. In particular, the proposed method randomly update sparse subsets of network parameters at each time step. With randomized sparse updates, overfitting problem of sequential-in-time methods can be mitigated, leading to better accuracy at test times. Moreover, the computational costs of training is reduced without losing expressiveness of the model, while accelerating the training process. Proposed method is validated on a wide range of evolution equations.

STRENGTHS:
* The paper is well organized and easy to follow. It also has a strong motivation inspired by the Neural Galerkin schemes and neural network dropouts.

* The paper has a strong theoretical grounds and corresponding toy examples to support the claim. For instance, two challenges of previous sequential-in-time training are well addressed in the section 2.2 with adequate examples. 

* The paper presents a wide range of ablation studies and various experimental settings to show the effectiveness of the proposed method. Most of the claimed benefits of the proposed method are validated empirically, including lower error rates and faster running time. 

* The proposed method is considered a big contribution for the field of sequential-in-time training methods.

WEAKNESSES:
* The performance of the model is only validated in standard benchmark equations. Validation in high-dimensional PDE problems can benefit the paper.

QUESTIONS:
* The proposed method is inspired by dropout in neural networks. Thus, I am curious about comparison results with PINN and CausalPINN, with different rates of dropouts applied.

LIMITATIONS:
Limitations are addressed by the authors
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-08