Lie Symmetry Net: Preserving Conservation Laws in Modelling Financial Market Dynamics via  Differential Equations
Authors: Xuelian Jiang
Tongtian Zhu
Yingxiang Xu
Can Wang
Yeyu Zhang
Fengxiang He
Venue: Accepted by TMLR ()

Abstract: This paper employs a novel Lie symmetries-based framework to model the intrinsic symmetries within financial market. Specifically, we introduce Lie symmetry net (LSN), which characterises the Lie symmetries of the  differential equations (DE) estimating financial market dynamics, such as the Black-Scholes equation. To simulate these differential equations in a symmetry-aware manner, LSN incorporates a Lie symmetry risk derived from the conservation laws associated with the Lie symmetry operators of the target differential equations. This risk measures how well the Lie symmetries are realised and guides the training of LSN under the structural risk minimisation framework. Extensive numerical experiments demonstrate that LSN effectively realises the Lie symmetries and achieves an error reduction of more than one order of magnitude compared to state-of-the-art methods. The code is available at https://github.com/Jxl163/LSN_code.

Published in Transactions on Machine Learning Research (May/2025
Lie Symmetry Net: Preserving Conservation Laws in Mod- elling Financial Market Dynamics via Differential Equations
Xuelian Jiang
Banggl
School of Mathematics and Statistics Northeast Normal University
Tongtian Zhu
ruiden&izju.com edu.cn
College of Computer Science and Technology Zhejiang University
Yingxiang Xu'
yrou@nenu.educc
School of Mathematics and Statistics Northeast Normal University
Can Wang
moon@zju.edu.
College of Computer Science
Zhejiang Universily
Yeyu Zhang
zhangyeguu@mail whw.a:/mmmllssseeeeeeee
School of Mathematics Shanghai University of Finance and Economics
Fengxiang He
F. F.HeWed.ac uk
School of Informatics University of Edinburgh
Reviewed on OpenReview: https: openreview. net/ forum? id= odrkffop9yyn
Abstract
This paper employs a novel Lie symmetries-based framework to model the intrinsic sym- metries within financial market. Specifically, we introduce Lie symmetry net (LSN),, which characterises the Lie symmetries of the differential equations (DE) estimating financial market dynamics, such the Black-Scholes equation. To simulate these differential equa- lions in symmelry-aware manner, LSN incorporates Lie symmetry risk derived from the conservation laws associated with the Lie symmetry operators of the target differential equations. This risk measures how well the Lie symmetries are realised and guides the training of LSN under the structural risk minimisation framework. Extensive numerical experiments demonstrate that L.SN effectively realises the Lie symmetries and achieves an error reduction of more than one order of magnitude compared to state-of-the-art methods. The code available at https: http:/gttttbbboo//i/tteee LSN code
1
Introduction
A classic approach for modeling financial market dynamics is via stochastic differential equations (SDEs). Through the application of the Feynman-Kac formula (Del Moral & Del Moral 2004), these SDEs can be transformed into corresponding Partial Differontial Equations (PDEs), such as the Black-Scholes (BS) equation
Merton
1973
Black Scholes
1973 Goodman & Stampfli 2001
Rodrigo & Mamon, 2006). Traditionally,
*Corresponding author.

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numerical methods such as Finite Volume Method (FVM) (Valkov 2014 and B-spline collocation methods [Kadabbajoo et al. 2012: Huang & Cen 2014) are used to simulate these equations. In recent years, AT driven methodologies, exemplified by Physics Informed Neural Networks (PINNs) (Raissi et al. 2019), have emerged as a powerful solution for solving differential equations by utilizing collocation data, a set of scattered points within the solution domain, to fit. their dynamics.
A defining characteristic of SDEs is their "symmetry". A major family of mathematical tools to characterise
Gazizov & Ibragimov 1998). Lie symmetries preserve the structural properties of solutions, simplifying the process of
the symmetry are Lie symmetry groups (Edelstein & Govinder 2009
Paliathanasis et al. 2016
finding solutions and improving accuracy (Misawal 20011 Kozlov 2010
Marjanovic et al. 2015 Gacta 2017
Marjanovic & Solo 2018). This principle is also applicable to the Black-Scholes (BS) equation in finance
Rao et al. 2016). Our vision is that the Lie symmetries can represent some intrinsic symmetry in financial
markets, though in an abstract manner (Kozlov 2010).
However, Lie symmetries remain largely unexplored in current Al-driven DE solvers. Neglecting this symmetry may lead AI-driven approaches to learn solutions performing well on training collocation data, yet fail to satisfy inherent structural constraints. This is caused by the imbalance or other limitations, such as the low quality of collocation data, which may compromise the generalizability of the learned solver. As result, the performance of the learned solver on unseen data remains uncertain, as widely reported in the literature
(Cohen et al. 2021 Yang et al. 2023 Li et al. 2024).
This paper aims to answer the following fundamental question:
Could Lie symmetries facilitate Al-driven DE solvers in simulating financial market dynamics, and how?
Motivated by this question, we design Lie symmetry net (LSN), which enables the simulation of financial market dynamics while preserving Lie symmetries.
Similar to many symmetries in physics, the Lie symmetries can be transformed into conservation laws (Kara
& Mahomed 2002 Edelstein & Govinder 2009 Khalique & Motsepa 2018 Özkan et. al., 2020). Specifically, for the Black- Scholes, the conservation laws derived from Lie symmetries are
D + DDT 0,
where Dt, Dr represents the partial derivative with respect to time t or asset price x, and (T* represents the conservation vector subject to the symmetry condition (i.e., Lie point symmetry operator) G, such that the action of G on the conservation vector satisfies G(T*, = 0 (Kara & Mahomed, 2000 Edelstein & Govinder 2009).
In our LSN, we design a novel Lie conservation residual to quantify how well the Lie symmetries are realised on one specific point the collocation data space that comprises asset price and time. This Lie conservation residual then induces Lie symmetry risk that aggregates the residual over the collocation data space, and thus characterises how Lie symmetries are realised from global view. It is worth noting that this Lic symmetry risk depends on the specific conservation law, and thus the specific Lie symmetry operator. This Lie symmetry risk is then integrated with risk functions measuring how well the LSN fits the collocation data (Raissi et al. 2019 Xie et al. 2023), and formulates the structural risk of LSN. We can optimise the LSN under the structural risk minimisation (SRM) framework (Shawe- Taylor et al. 1998 to learn an differential equations solver while preserving the Lie symmetrics.
Extensive numerical experiments are conducted to verify the superiority of LSN. We compare LSN with state-of-the-art methods including IPINNs (Bai et al. 2022), sfPINNs (Wong et al. 2022). fPPNNN (Wong) ct al.] 2022 and LPS (Akhound-Sadegh et al.) 2024). The results demonstrate that LSN consistently outpertorms these methods, achieving error reductions of more than an order of magnit Specifically, the error magnitude with single operator reaches 10-3, while with combined operators (refer to Section 5.3 it further decreases to 10-4
The paper structured as follows. Section provides an overview of related work. Section 3 discusses the background of PINNs and SDFs. Section introduces the methodology of LSN. Section presents numerical
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experiments to validate the effectiveness of LSN. Finally, Section P for future research. Appendix A provides additional background, models and the theoretical analyses of LSN.
draws conclusions and outlines directions
2 Related Works
Numerical Equation Solvers. Numerical methods have long been essential for solving partial diffcrential equations in various domains, including financial market modeling. Significant, progress has been made in this area with models such as the Black-Scholes equation. Traditional approaches, including finite volume methods (FVM) (Valkov) 2014) and B-spline collocation methods (Kadalbajoo al. 2012 Huang & Cen 2014), have been widely applied to solve these equations {Koc et al. 2003 Rao et al., 2016). These grid-based techniques rely on discretizing the spatial and temporal domains, translorming the continuous equations into discrete problems suitable for simulation. However, these methods often come with high computational complexity, which may limit their applicability.
Neural Equation Solvers. In recent years, there has been a gradual increase in applying neural networks to solve differential equations. Two main approaches have emerged in this arca. The first onc, neural operator methods (Li et al., 2020 Lu et al. 2021; Kovachki et al., 2023 Hao et al., 2023), focuses on learning the mapping between the imput and output functions of the target equations. In contrast, the second approach, Physics-Informed Neural Networks (PINNs) (Raissi et al.) 2019), directly approximates the solution of the equations, rather than relying exclusively on the values of the function derived from collocation data. PINNs and their variants, such as sIPINNs (Wong et. al., 2022) and ITPINNs (Wong et al. 2022), have gained popularity for utilizing physical laws into the training process. Recent studies have successfully applied PLNNs to solve financial equations, introducing efficient methods like IPINNs (Bai et al., 2022). which incorporate regularization terms for slope recovery.
Input Space Symmetries. Despite significant advancements in machine learning, the inherent symmetries within input data remain underutilized. Equivariant Neural Networks (ENNs) represent an innovative architectural approach explicitly designed to encode and utilize these symmetries, resulting in the property
of equivariance (Celledoni et al. 2021 Satorras et al. 2021 Gerken et al. 2023) This property enables ENNs to efficiently model data with structured invariances. In the domain of computer vision, ENNs have shown remarkable efficacy over traditional neural networks. For example, in image classification tasks, they achieve higher accuracy in recognizing objects subjected to transformations such as rotation or translation RRoas-GGomz et al. 2024). By preserving the symmetrics of input data, ENNs retain critical geometric information during feature extraction (He el al.] 2021]. Moreover, ENNs also exhibit exceptional performance in physical simulations by effectively modeling symmetrical transformations of physical systems (Bogatskiy
ct al. 2024); thereby improving both the accuracy and computational efficiency of simulations.
Parameter Space Symmetries. Over recent decades, a range of studies have analyzed symmetries in neural network parameter spaces transformations of network parameters that leave the underlying network function unchanged (Hecht-Nielsen)
1990 Sussmann 1992). Notable examples of such symmetrics include invertible linear transformations in linear networks and rescaling transformations in homogeneous networks
[Bariiarayanan et al. 2015 Du et al. 2018 Recent studies further provide deeper insights and expand the
scope of parameter space symmetries. Zhao et al. (2023) investigate diverse parameter space symmetries
and derive novel, nonlinear, data-dependent symmetries. Ziyin (2024) establish a systematic framework for analyzing symmetries, showing that rescaling symmetry induces sparsity, rotation symmetry leads to low-rank structures, and permutation symmetry facilitates homogencous enscmbling. Further, Ziyin et al. (2024) examines how exponential symmetries, a broad subclass of continuous symmetries, interplay with stochastic gradient descent.
Lie symmetrics. While input space and paramcter space symmetries have played crucial roles in neural network design, they have rarely been applied to characterize the intrinsic symmetries of differential equations. Lie symmetry analysis, in contrast, provides a systematic framework for analyzing these symmetries. It represents a significant type of symmetry with widespread applications in mathematics (Olver) 1993 Tbragimov] 1995). As a powerful tool for solving partial differential equations (PDEs), Lie symmetry method leverages the symmetry group of an equation to reduce its order or transform it into a simpler, more tractable form,
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Despite its well-established theoretical foundation in traditional mathematical fields, its integration into neural DE solvers remains largely unexplored.
thereby simplifying the solution process (Gazizov & Ibragimov 1998 Liu et al. 2009 Oliveri 2010
A more recent work by Akhhund-Sadegll et al. 2024 proposes to incorporate Lie symmetries into PINNs by minimizing the residual of the determining equations of Lie symmetries. While this approach offers an interesting direction, our LSN follows different methodology. Specifically, their Lie point symmetry (LPS) method focuses on minimizing these symmetry residuals, whereas our LSN realises Lie symmetries by preserving the conservation laws derived from the Lie symmetry operators. These conservation laws are fundamental principles inherent to the system described by the diffcrential equations. Additionally, LPS has been validated only on the Poisson and Burgers equations in their original paper and its effectiveness in leveraging inherent symmetries in financial markets remains unclear. In contrast, our comprehensive comparative experiments financial domain, specifically on the BS equation across various parameters, clearly demonstrate the superiority of LSN over LPS by reducing testing crror by an order of magnitude.
3 Preliminaries
This section provides the essential background knowledge. We begin with an introduction to Physics-Informed Neural Networks. We then cover stochastic differential equations and explain how the Feynman- -acc formula allows for the transformation of a SDE into a corresponding PDE. To concretize this theoretical framework, we provide illustrative examples including the Black-Scholes equation. For further terminology and models related to finance and Lie symmetries, such as the Vašiček equation, please refer to Appendix A
3.1 Physics-Informed Informed Neural Networks (PINNs)
PINNs integrate physical information from the equations to approximate numerical solutions of PDEs, rather than relying exclusively on collocation data. The use of PINNs solve differential equations typically begins N. with generating a collocation dataset S S= {{{((t [(x). 4)), N. {(TT,1")) N. [-r],, ). f by randomly sampling points
within the solution domain. To understand how PINNs operate, consider the following evolution partial differential equation,
@u(ar,) = L[u] ot u(x, 0) = p(x)
V(x,1) E 2 [0,T],
V Q Y(y,t) 02 [0,
(1)
u(y, t) tyy,t)
where L[u] represents a differential operator, Ω denotes a bounded domain, (px)) and (yy, f) correspond to the initial and boundary conditions, respectively, T refers to the terminal time, and u(z, t) is the function to be determined. To solve Equation (1. PINNs approximate the exact solution by modeling u as a neural network à and minimizing an empirical loss function.
CIINN=
1 N
Via dia(FF 47) Ot
[[0] (
1 No
Σ
(2 (g) -(((z N₂ Σ (0,00 -(()))
where denotes the work parameters. The first term evaluates the residual of the PDE, while the subsequent terms quantify the errors associated with the boundary and initial conditions, respectively.
3.2
Stochastic Differential Equation (SDE)
SDE [Oksendal & Øksendal 2003 provide a mathematical framework for modeling systems inffuenced by random disturbances. To understand the dynamics of a stochastic process X,, we consider the SDE of the following general form
dx, na( (x,,, t)dt + (X,, f)dW,
(2)
where X, represents the stochastic variable of interest and W, standard Wiener process (as defined Definition A.I.p The functions p(X,,t) t) and oX,,, t), known as the drift and diffusion coefficients, respectively, are functions that characterise the deterministic and stochastic components of the dynamics.
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The Feynman-Kac formula provides a critical theoretical framework to establish connection between certain types of PDEs and SDFs (see Definition A.6 We illustrate this with representative example from finance.
Example (Black-Scholes equation (Janson & Tysk 2006)). Considering a frictionless and arbitrage-free financial market comprising a risk-free asset and a unit risky asset, the dynamics of the market can be modeled by the following SDF:
dæ = rx,dt + aa,dW,,
(3)
where x denotes the price of unit risky asset, L represent time, is the volatility, is the risk-free interest rate and We is a standard Wiener process (refer to Definition A.1 Applying the Feynman-Kat formula, the Baack-Scholes equation for evaluating the price u(æ, t) of European call option (refer to Definition A.2) is derived as follows:
{
au + 1'22220'' + rI Ju -r rr== àt u(x, T max( K,0)
Ω x [0,T], T,
dx²
0z
=
0n [].
where K is the strike price, T is the expiry time of the contract and bounded domain. The solution this equation can be written as follows [aai et al., 2022
u(x, = xNd()) exp r(T- ((d)) (r (r
d2 = di - -Vo---T-
OT
where N denotes the standard normal distribution.
3.3 Lie Group Analysis
Groups, which mathematically characterize symmetries, describe transformations that preserve certain invariances. Formally, a group (G,-) is defined set G equipped with binary operation that satisfies the properties of associativity, contains identity element E G, and ensures the existence of an inverse element 9 for each E G (Rotman 2012). When groups are equipped with the structure of differentiable manifolds, they are referred to as Lie groups and play a fundamental role in the analysis of continuous symmetries. (Knapp & Knapp 1996). Lie group analysis provides a powerful tool for studying symmetry, conservation laws, and dynamic systems of equations (Gazizov & Ibragimov 1998 Paliat al. 2016
Brandstetter al. 2022; Otto al. 2023 Dalton et al.) 2024 Yang al. 2024). The goal of Lie group analysis to identify the symmetries of an equation, especially those transformations under Lie group actions that leave the equation invariant. These symmetries facilitates conservation law derivation EEdelteei &
Govinder 2009). simplifies solutions, and lowers computational complexity (Rao et al. 2016
Khalique &
Molsepa 2018
4 Lie Symmetry Net
In this section, we introduce Lie symmetry net (LSN). In Section 4.1 we briefly derive the corresponding conservation law from the Lie symmetry operators of the target equations, which in turn lead to the Lie symmetry risk of LSN. In Section 4.2 discuss the structure risk minimization of Lie Symmetry Net based on the Lie symmetry risk.
4.1 Lie Symmetries in Equations
This subsection presents the Lie symmetry operators for the Black-Scholes equation, and derive the corre- sponding conservation laws, which allows us to define the associated Lie symmetry risk.
Lie Symmetry Operator. Lie symmetry operator is major mathematical tool for characterizing the symmetry in PDFs (see Definition
A.3 (Paliathanasis et al. 2016 The Lie symmetry operators (Gazizov &

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6g t0 Lyou + Leo + lzc
C= +42
Hypothesis space of neural
62 DT D.T
network solution
Opnaa
Conservation Law (ū,xtt (ū,
u
Lie Symmetry Operator
The incorporation of Lie symmotries blocks into PINN architecture. Right: The hypothesis space of PINN, denoted âPTNN (blue circle), is subset of the broader neural network solution space ( gray circle). u" represents the exact solution. The yellow circle signifies the hypothesis space that satisfies symmetry conditions. The incorporation of Lie symmetries blocks produces our solution Hours. In simpler terms, being closer to the space where the exact solution exists leads to fewer errors. The non-interscction between "ŪPINN' and "ūours" in the figure is a specific case provided to enhance clarity in our exposition.
Figure Schematic Diagram of Lie Symmetry Network Architecture. Left:
Tbragimov
1998
Edelstein & Govinder
2009 of BS Equation are given by the vector field
0
θ
G. o(1, x)
Gi
G2
G3 Ou'
Ou'
0t.
dr
θ
0
θ
+ (In Zt)u a
G4 2t 0t
+
(Inx
Zt)
2rtu
G5 0² tæ Or
(5)
dr
Ou'
Ou'
①
Gg 20 22
+ 20 x dr
tx In
+ ((n Zt) 2202 - -o²t)
Ou''
where Z= 02/2, and @(,,)) is arbitrary solution to Equation 4 without any boundary condition initial condition. The first symmetry Gg is an lincarity. These Lie symmetry operators form an infinito Lic group vector space (Edelstein Govinder 2009). The derivation of Lie symmetry operators for general differential equations included in Appendix A.3
symmetry, arising as a consequence of
These Lie symmetry operators not only provide deeper insight into the structure of the PDEs but also form the foundation for deriving conservation laws associated with these equations.
Conservation Law. Similar to many symmetries in physics, the Lie symmetries can be transformed to conservation laws (ara & Mahomed 2002 Edelstein & Govinder 2009 Khalique & Motsopa 2018 Özkan et al. 2020). In this paper, we interpret the Lie symmetry point operators as the following conservation tion laws: regardless how the space x, time t, and exact solution u vary, the conservation vector (T*,TT) TT) corresponding to the Lie point symmetry remains zero, i.e.,
(u, =0
where the Dr, Dy represents the partial derivative with respect to time / or space H, and (T', (", represents the conservation vector subject to the symmetry condition (i.e., Lie point symmetry operator) G, such that the action of G on the conservation vector satisfies (") = (Edelstein & Govindor 2009). is worth noting that this Lie symmetry risk depends on the specific conservation law. The derivation of the conservation law for general differential equations is included in Appendix A.4

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We can derive the conservation law of the operator G2 (see Equation 5 of BS equation follows & Govinder 2009 :
[Edelstein
Ou (t) + d
+ 2Bu
T (u, x, x))=
σ²x e e-r
dx
I
du ((t) + u al(
Ou dz
2ru o2x
T (u, 1,
Ōt + g(t)
Bue rt
+ $
ze
0t
where a and H are arbitrary constants, and z(t) and g(t) are arbitrary functions with respect to t. Unless stated otherwise, consider 1,
1, and g(t) = 1. The selection of G2 is primarily motivated by its simplicity, as effectively illustrates the fundamental concepts of the proposed method. Additional experiments in Section 5.3 demonstrate the effectiveness of the proposed methods when applied with various operators and their combinations the Vašiček equation.
= [(1)
We can then define the Lie conservation residual RLie according to Equation G to evaluate the extent to which the Lie symmetries are realised at specific point in the collocation data space.
Definition 4.1 (Lic conservation rosidual). Combining the Lie symmetry operator and the conservation law, we define the Lie conservation residual of â as follows:
R [4] = (1) + (
(8)
where the D., Da represents the partial derivative with respect to t or x, and (T', represents the conservation vector subject to the symmetry condition G.
We can aggregate the Lie symmetries residuals over the entire collocation data space to obtain the Lie symmetry risk, which characterises the degree to which the Lie symmetries are realised from a global perspective.
Definition 4.2 (Lic symmetry risk). According to the expression Equation the definition of Lie symmetry risk is provided as follows:
/
Lrie(0o) (I,
J R [āo] (x,, 1)1² dxdl,
where dpp denotes the network output, and 0 denotes the network parameters.
Remark 4.3. The Lie symmetry risk Liie focuses solely on learning the symmetry of the problem without taking into account the underlying physical laws of the problem.
This Lie symmetry risk is defined over the collocation data distribution, which is unknown in practice. We thus define the Empirical Lie symmetry risk Liee as an approximation of the Lie symmetry risk L
Definition 4.4 (Empirical Lie symmetry risk). Summing up the Lie symmetry operators N; discrete points provides approximation to the Lie symmetry risk.
(1,e
N Σ RLie [o](<)7{{))'
(9)
where S = {[p) 1
represents the set of these N; discrete points.
4.2 Structure Risk Minimisation
In this subsection, present the structure risk minimization of LSN based on the Lie symmetry risk. For
clarity, we reformulate the Black-Scholes Equation { to match the format of Equation [ (see Appendix A.2
aligning it with the Vašíček Equation 23 in the Appendix A.3 for a coherent presentation, where
aa)) Pu(a,t)
Ou(x,t) dx
L/u := 2
+ +(x)
+ v(z)u(z,t),
(10)
dx
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is differential operator with respect to three bounded affine functions (xx), jl(z) and e(x)) (for BS equation: (z) dæ, (x) = rx, v(x) = -r and (p(x) maa K, K, (for Vašiček Equation (23) a (xx) v 2ox 0, p(z) X(3 -3 = = v(r)
and (xx) = 1).
Data Fitting Residuals. The following functions R2 (j = {i, s. t)) characterise how well the LSN is fitting the collocation data according to Equation 1 for vú E C* (TR.)
[x 1) 86(a.1) [[ū 1) R. (y. t) t(y,t) tly.t)
(x, 1) x [0,7], (y,t) 02 x [0,T], xEf.
(11)
(x) = a(0, x)) p(x)
We define the empirical loss function Ê1 to approximate the population risk C1 in fitting the collocation data, by utilizing the aforementioned residuals, as follows:
L [ol tx, LPDE(LS) (io) (x.t) Lec (ão] (z, t)
Ciclãol (2,,)
N: (R] 42)2
N.
N; Σ N
RR [ (x², (2)
[R (x,))²
N₂
N
Here, 8 8= = { [aaf, (a),,,))),, (()),)))) N. {xT)) denotes the collocation data set, which is generated by sampling points within the domain Ω x [0, T] according to a Gaussian distribution.
Remark 4.5. & essentially corresponds to the Plysics-Informed Neural Networks (PINNs) (please refer Section 3 which accurately estimates the majority of the training collocation data based on the inherent. physical laws of the problem. However, it does not explicitly consider the symmetry within.
Structural Risk of LSN. We next present an empirical approximation of the structural risk E() of LSN.
Definition 4.6 (Empirical structural risk of LSN). The empirical loss of LSN is defined as follows
(00,8):=)&PDD((.. (xC + XaL1o(0, (.C
(12)
No } are the training collocation data sets, and A where S = S ((,,,))); S. ((2,, S. (x (1 = 1, 1,....4) denote the hyperparameters.
We train LSN by solving the following minimisation problem,
0r arg min (L (22 + XzLrc(0,5.) XeLisc (0,
and the minimum åg. corresponds to the well-trained LSN.
5 Experiments
In this section, we conduct three main experiments. In Section 5.1 we perform an ablation study on LSN, comparing it to PINNs (Raissi et al. 2019) under varying equation parameters. Next, we compare LSN with state-of-the-art baselines, including IPINNs (Bai et al. 2022). siPINNs (Wong et al., 2022 ffPINNs (Wong lt al. ), and LPS (Akhound-Sadeghh al. 2024), thereby further validating its superiority. In Section 5.2 extend LSN to multiple models, providing experimental comparisons under both single-operator and combined-operator settings to verify its effectiveness. In Section 5.3 we test LSN real financial data from Yahoo Finance to demonstrate its practical applicability.
We start by providing a brief introduction to the parameter settings for all experiments, with specific parameters for each experiment detailed in their respective sections.
Data. The small-scale experiments employ a training set 50 internally scattered points and 2000 points randomly placed at the boundaries, while the test set consists of 2,500 (or 200) uniformly sampled points. The
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Table 1: Experimental Parameter Configuration. Values parentheses represent weights for the enlarged collocation data set, while values not parentheses represent weights for the small collocation data set.
RFR (T)
volatility
weight
learning rate (Ir)
iteration
(o)
X
A2
X
A
0.1
0.4
0.001 (0.001)
1(1)
(01
1(0.1)
0.001
200,000
0.1
0.5
(.001(0.0001)
1(1)
0.1(0.1)
10(0.01)
0.001
200,000
0.11
0.4
(0.01(0.001)
1(1)
0.1(0.1)
1(1)
0.001
200,000
large-scale experiments employ training set of 2000 internally scattered points and 8000 points randomly placed at the boundaries, while the test set consists of 2,500 (or 200) uniformly sampled points.
Neural Architecture and Optimiser. The LSN network employs fully connected architecture, consisting of 9 layers with each layer having width of 50 neurons. The tanh function is used as the activation function. For optimization, we choose Adam with an initial learning rate of 0.001 and a learning rate decay factor r.
Equation Parameters. For the parameters of the Black-Scholes equation, we set them to K = 10, x E [0,20], and / [0, 1], following the conventions established in the existing literature (Bai el al. 2022).
Evaluation. The evaluation metrics include relative test error (refer to the definition in Section Defini- tion A.5 and conservation error Lie.
5.1 Experiments on Sackl-Scholes Equation
In this section, we perform experiments focused on the Black-Scholes (BS) equation to conduct ablation studies and comparative analyses LSN.. In Section 5.1.1 we perform ablation studies on LSN with both small- and large-scale datasets, comparing it to the baseline PINN method, to demonstrate its performance improvements across varying data scales. In Section 5.1.2 we compare LSN with several statc-of-the-at methods, including the Fourier frequency-based IPPINN and sfPINN, the BS-equaaiion-speciie IPINN, and the Lie symmetry-based LPS, to comprehensively assess its performance.
5.1.1 Comparison with PINNs
Experimental Design. We conduct comparative experiments between LSN and baseline PINNs under the following four sets of hyperparameter setups.
For the first configuration, we chose r= 0.1, o = 0.05, a learning rate decay rate = 0.99, and Iterations = 50, 000, the weight for LSN's loss function was set as \ 0.001, A \4 = 0.001, while for PINNs, the weight was set as λ 0.001, A2 0.999 and A3 0.001. For the second configuration, select 0.1, 0.2, learning rate decay rate of r 0.95, and Iterations 90, 000. The weights for the loss functions remains the same the first configuration. For the third setting, the experimental parameter settings for the small collocation dataset under other parameters are given in the Table [ Regarding the fourth setting, the experimental parameter settings for the enlarged collocation data set are provided in the Table [ with values in parentheses.
A 0.999, X 0.001 and
We visualize the numerical solutions obtained on the test set using these two sets of hyperparameters Figure 2 In identical experimental configurations, LSN outperforms PINNs, achieving point-wise error magnitude of 10-2 in contrast to 10-1 observed with PINNs
The error curves for LSN and PINNs with respect to the number of training steps for the third set of parameters are presented in Figure 3 It can be observed from Figure 3 that the conservation error of PINNs is the order 10-1, whereas LSN can achieve an error on the magnitude 10-1 Furthermore, the relative error of LSN also consistently remains lower than that of PINNs.
In the fourth set of experiments, we increase the number of collocation data points to 10k. It can be observed from Figure that increasing the number of collocation data points improves the accuracy of both PINNs

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God (wttEN
E0000
0M
Figure 2: Visual representations of numerical solutions obtained using LSN and PINNs, along with absolute
errors compared to the exact solution. The configuration of parameters is as follows: (1) r = 0.1, o = 0.05,
I = 0.99, and Iterations = 50, 000 (the left two column); (2) r 0.1, o = 0.2, r 0.95, and Iterations = 90, 000 (the right column).
Logiing px0.4)
Logtng plrt ex88)
Log-lng gwd.4)
PNN
RMN
Logeing plet (r 0.0.))
Loging plt (ru 6.6.5)
Logelgg plat (01.11111111111 0..4
Cres wat ever
Figure 3: Log-log relative test error curves of PINNs and LSN under the third parameters configuration.
Table 2: Comparisons of relative test error of LSN and PINNs after 80,000 training iterations. Here r represents the rate of learning rate decay, while "Factor' represents the ratio of the test error of PINNs to that of LSN.
RFR (r)
Volatility (r)
Relative test error
Factor
PINNs
LSN
0,1(1 = 0.99)
0.05
3.1 x 10-3
4.5 x 10-4
6.9
0.1(T 0.95)
0.05
1.1 10-3
5.4 x 10-4
2.0
0.1 0.95)
0.2
5,5 10
1.6 10
3.4
0.1(T 0.95)
0.4
3,5 10
1,3 x 10
2,6
0.1 0.95)
0.5
8.0 10
2.4 x 10
3.3
0.11(T 0.95)
0.4
3.0
10
1.1 x 10
4.4
and LSN. Notably, the parameters r 0.11, 0.4, the test error magnitude of PINNs and reaches 10 and 10-3, respectively.
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Logilg plet frud.b, 0x0-5)
Legiog plet frub. ex
Loging pist (mu (m-6.33, m-4.4)
PNN
L5N
L5h
L5h
14000 Lrer
Raanng Kowe
Hesstee
Log teg piot (owd. awb-))
Log tog part (ow0.l, owb
Log tog piat (rw0. 11, gw0.4)
PNN
PNN
HMN
LSA
15h
15h
Cossinattinn 2r1
Messes
Figure 4: Log-log relative test error curves of PINNs and LSN under the forth parameters configuration.
Table 3: Comparison of cumulative training time for LSN and PINNs to achieve different error thresholds.
Error Threshold
Cumulative Training Time (s)
LSN
PINN
7.00
x 10-3
8.85 x 102
1.99 x 10a
6.00
x 10-3
8.92 x 102
2.26 103
5.00
x 10-3
9.99 x 10a
2.60 x 103
4.00
x 10-3
1.11 x 10a
3.25 x 10
3.00
x 10-3
1.35 x 10
4.44 x 10
To provide a more intuitive demonstration of the superiority of LSN, consider the test accuracy of the method under different equation parameters, shown in Table 2 Compared with vanilla PINNs, LSN can reduce the relative test error by up to 7 times, with an average improvement of 2-4 times.
In addition, we conducted additional experiments on training efficiency under the parameter settings r = 0.2 and a = 0.2. In these experiments, we compared the cumulative training time required for LSN and PINNs to achieve specified error thresholds. The results are summarized in Table 3 As obscrved, LSN exhibited a consistent and significant efficiency advantage across all tested error thresholds. For instance, al an error threshold of 7 x 10-3, LSN achieved the target precision in 8.85 x 102 seconds, whereas PINNs required 1.99 x 103 seconds. Despite the addition of regularization terms in LSN, which complicates the loss function to be optimized, the experimental results indicate that LSN does not converge more slowly than PINNs in terms of wall-clock time and simultaneously achieves higher accuracy.
5.1.2 Comparison with State-of-the-ee Methods
We conduct comparative experiments between LSN and several state-of of the-art methods including IPINNs BBai et al. ) sfPINNs (Wong al.] 2022). fPPINNs (Wong et al.] (2022) and LPS (Akhound-Sadegh et al. 2024), under different equation parameter setups (i.e., different risk-free rate and volatility), following
Ankudinova & Ehrhardt (2008); Shinde & Takale 2002 Bai et al. 2022)).
Experimental Design. All methods share the following hyperparameter setup, i.c., learning rate ir = 0.001 and learning rate decay rate r = 0.95. The training steps are set as 80,000 and 200,000, depending on the speed of convergence.
Figure 5 provides Log-log relative test error curves and function approximation results of LSN, PINNs, and PINNs variants. The results demonstrate that the relative error of LSN consistently remains below those of vanilla PINNs and their variants across different experimental settings. Both s(PINNs and fiPINNs
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Loghag 1, (5)
di 66,7-0
Leglogr-
F-Ele 2-ELe-02,t00 02,1-0200
VMM
Roat
Deat
1400 Enge
10X
Banssion
Figure 5: Log-log relative Lest error curves and function approximation results of LSN, PINNs, and PINNs variants. The error curves are shown in the first and third rows, while the function approximation results at = 0.30 are presented in the second and fourth rows.
Log-leg piet (rwd. ow0.4)
Lop pirt (owo ex0.)
Log-log plat freO, owo
PINN
PNNN
HNN
Logilgg piet (rnd.,, e.0.4)
Loplng plot en0.1
e10.1
Lag-ngg fra.1.. am0.4)
LSh
LSK
(P5
LPS
Barstinn
harddos
Figure 6: Log-log relative test error curves of PINNs, LSN and LPS under the third parameters configuration.
exhibit unsatisfactory performance under certain parameters, occasionally performing even worse than vanilla PINNs. This underperfommanee might be attributed to the fact that sfPINNs and ffPINNs are more suited to scenarios with sinusoidal-form solutions, thereby failing to effectively approximate the complex solution of the Black-Scholes equation (Wong et al., 2022
We conduct comparative experiments among LSN, LPS, and PINNs using the same weights, as shown in Figure 6 Specifically, LSN and LPS share the same weights A; for i = 1,...,4, while PINNs share the same weights X; for i i= 1, ,,...,3 as LSN and LPS but with \4 = O. The experiments demonstrate that LSN outperforms both PINNs and I.PS. Additionally, LPS exhibits overall superior performance compared to PINNs when early stopping is employed.
For more fine-grained comparison between LPS and LSN, we further finetune the weights 4 (i= 1, (i=1,... 4) of the loss function of LPS under different configurations. Notably, 4 (i=11 1,...)..,33) in LPS serve the same purpose as A; (i = 3) in LSN, while 4 in LPS determines the weight of the symmetry residuals, and xa in LSN determines the weight of the residuals of the conservation laws corresponding to the Lie symmetry operators. To illustrate the specific process of weight tuning for LPS, consider the example with r = 0.1 and a = 0.4, as shown in Figure 7 We start by fixing all weights to 1 and then traverse I values from 1,0, 0.01, 0.001, 0.0001] in descending order, selecting the best value of 1. Similarly, we traverse la values and find that le performs well in the range of 0.1 to 10. We then further subdivide this range into 2, 1, 0.5, 0.25, 0.1] for experinentation and select the best value for 12, which is fixed thereafter. This process is repeated for finetuning other parameters of LPS.
After finetuning the weights of the loss function of LPS, and using the previously set weights for LSN and PINNs, we validate the performance of LSN, PINNs, and the finctuned LPS model. As shown in Figure R after extensive weight tuning, LPS outperforms PINNs with carly stopping in relative test error but remains surpassed by LSN in terms of relative test error and conservation law error.
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PMN
PRN
FRNN
PAN
1P5,1.30 10
1P5,1_5.3
LP5 1P5,0,10
LFS, 4
1P5.1.1
1P5,1 5.25
UP5,0_1
4,1
US
(P5,R_65
(PS_0_93
4,011
[P5
(P5_0.9.01
LPS, 4
1P51110.031
(P5, 0_9.000
LPS, 4, 4,0.00
UPS_ 3P5,0,000051
UPS. 4,0.00
10
Figure 7: Log-log relative test error curves of PINNs and LPS.
Leg log piet trwo o-0.4)
Lop plot (rwD e-051
Loging plat frwo ow0.4)
PNN
L.5
15h
15h
175
UPS
US
Serston
Log-top piet towo. ow0.4)
Lop pl1t tow0. e=b.)
log pirt (rw0.11, awa.4)
PINN
PNNN
HMN
LSN
15h
15h.
Greanvtton Eree
LP5
1P5
UPS
Barstin
herados
Figure 8: Log-log Error Curves Over Training Steps: PINNs vs. LSN LPS, After individually tuning the weights for LPS, the performance of LSN is evaluated on the testing collocation data set using two metrics.
This obsorvation can be explained by the fact that LSN potentially retains a broader spectrum of symmetrics than LPS by leveraging conservation law systems. These systems incorporate additional symmetries, including potential symmetries (Bluman 1993 Pucci & Saccomandi, 1993), nonlocal symmetries (Bluman et al. 1988 Olver 1993), and novel local symmetrics (Edelstein & Govinder 2009). By utilizing the system of conservation laws, the symmetry analysis of the original equation is extended, thus revealing previously unexplored structures and solutions.
We validate this claim with the following experiments. For the equation parameters r = 0.1 and a = 0.4 as shown in Figure g we conducted experiments with 100k training steps. The Lie symmetry operator G2 is unique, while the derived conservation laws not, they involve freely chosen functions 1(t) and g(t) (sce Equation ( We defined two conservation law operators: LSN: O with l(t) = and g(t) = 82, and LSN:
O2 with ((t) sin(t) and g(t) = cos(t). We also combined these operators as LSN: O, + + As shown in Figure 9 upon convergence, both LSN: O₁ and LSN: O₂ demonstrate superior performance compared to PINN and LPS, with the combined operator LSN: O1 + O2 achicving the highest accuracy. These results indicate that the flexibility in choosing 1(t) and g(() allows the conservation laws to integrate additional information, thereby enhancing solution accuracy.
5.2 Experiments on Other Differential Equations
This soction illustrates the methodological flexibility of LSN when applicd to the Vašičok model, the Maxwellian tails model, and the KdV model. In particular, experimental results on the Vašišek model with different operator combinations indicate that stacking multiple operators can potentially improve accuracy.
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Log-log plot (r=0.1, a=0.4)
PINN
LSN:O:
LSN:O>
Relative Error
LSN:O; + O2
LPS
10-1
Figure 9: Log-log Error Curves Over Training Steps: PINNS vs. LSN vs. LPS. For O1, the functions are defined as ((t) t and (() = 8 (see Eq. (7)). For O2, ((t) sin(t) and g(t) cos(t). The resulting operator O₂ + O2 represents the combined effect through linear superposition.
Vašiček Model
Vašiček Model
18
16
15Nc
L5No
LSNs
LSN
Conservatoon
Reletive
19-1
Auso
Meration
Iteraiio
Vašlček Model
(b) Vašiček Model
KdV Model
Maxwellien Tails Model
Relebiee Eiros
Relaaive Error
12-1.
10
Beration
Iteration
(c) KdV Model
(d) Maxwellian Tails Model
Figure 10: Log-log Error Curves Over Training Steps: PINNs vs. LSN vs. LPS. The first row denotes: For the Vašiček equation, Gg (Gs) denotes the use of single symmetry operator corresponding the conservation law, while G5 Ge represents the linear combination of two operators. The second denotes: The algorithms are further extended to the KdV model and the Maxwellian model.
5.2.1 Vašiček Equation
This section extends the LSN algorithm to the Vašiček equation (Privault 2022), where experiments are performed using both individual operators and stacked combinations. detailed description of the Vašiček model provided in Appendix A.2
Experimental Design. In this experiment, the parameters the Vaštčck model are as follows: 0.03, 3 = 0.08, 7 -1, 0.03, Ω and T The dataset consists of 500 interior points and 200 boundary
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NDX European Call
1750
Market Price
LSN Price
1250
1000
Price
750-
6
10
Figure 11: Predicting European call options on Nasdaq 100 index using real financial data with LSN.
points. The network is designed with two layers, each containing 10 neurons. The training is performed for 100,000 iterations with a learning rate of 0.001 and learning rate decay factor of T 0.95.
As shown in Figure 10(a) and 10(b) extending LSN to the Vašiček equation further illustrates its ability to handle diverse operator structures. It is observed that while the use of single operator already achieves significant improvements over PINNs, the performance gains achieved through combined operators are even more substantial. This demonstrates the flexibility of our method: it can effectively utilize both single operators and combinations of multiple operators.
5.2.2 KdV Model
This section applies the LSN algorithm to the KdV equation (Iraagimvv 2007], with experiments conducted using a single operator. The KdV equation, a nonlinear partial differential equation commonly used in fluid mechanics and related fields, expressed as u4 = Uzax
Mig.-
Experimental Design. The KdV model parameters are configured follows: the spatial domain is Ω = [0, and the temporal domain is T 1. The dataset includes 100 interior points and 400 boundary points. The neural network architecture consists of 4 layers, each containing 50 neurons. Training is conducted for 200,000 itcrations with learning rate of Ir = 0.001 and learning rate decay factor of r = 0.95.
In the experiment using a single operator (see Figure 10(c) LSN achieves a relative test error on the order of 10-4, This result is significantly better than the PINNs method (accuracy on the order of 10-2) and the LPS mothod (aceuracy on the order of 10 10-3), demonstrating LSN's superior accuracy and stability in solving the KdV equation.
5.2.3 Maxwellian Tails Model
This section applics the LSN algorithm to the Maxwellian tails model. The Maxwellian tails model used to describe the behavior of particles in system where the distribution of velocities follows Maxwellian distribution, with particular focus the high-energy tails of this distribution, which is expressed Unt + u₂ + n² 0.
Experimental Design. In this experiment, the parameters of the Maxwellian tails model are set as follows: the spatial domain is 52 = [1, and the temporal domain is T 1. The dataset includes 100 interior points and 100 boundary points. The neural network architecture is designed with 4 layers, each containing 50 neurons. Training is performed for 200,000 iterations with a learning rate of Ir 0.001 and a learning rate decay factor of r = 0.95.
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In the experiment (as shown in Figure [0(d) , the decreasing relative error over training steps shows that LSN consistently outperforms LPS and PINNs. This result demonstrates the superior performance of LSN in the Maxwellian tails model.
5.3 Experiments on Real Market Data
In this section, extend the experiments to real-time market data. The focus is European call options based on the Nasdaq 100 index. The neural network used for option pricing relies two fixed inputs: volatility (a) and risk-free interest rate (r). In the experimental setup, the implied volatility from the OptionMetrics dataset (Wachowicz) 2020) is used. For the risk-free interest rate, the average yield of one-year Treasury bonds during the option period taken as the average yield of one-year Treasury bonds during the option period, obtained by scraping data from Yahoo Finance. We train a LSN model for this type of option and defines the required range of spot prices. During inference, these independent models are used to predict market prices.
As shown in Figure 11 LSN performs well (Dhiman & Hu 2023), but some discrepancies remain compared to market prices, potentially attributable to market supply and demand dynamics or other external influences. However, overall, the LSN method performs well, and a high correlation is observed bet ween market prices and predicted prices.
6 Conclusion
This paper proposes Lie Symmetry Net (LSN) to solve differential equations for modeling financial market dynamics by exploiting the intrinsic symmetry of collocation ion data. The Lie symmelries of these equations is interpreted as several conservation laws. We define a Lie symmetries residual to measure how well these conservation laws are realisod at specific points in the collocation data space, which then integrated over the entire collocation data space to form a Lie symmetry risk. LSN is optimized under structural risk minimization framework to balance the Lie symmetry risk and the original collocation data fitting residuals. Extensive experiments demonstrate the effectiveness and scalability of LSN, showing that the test error is reduced by more than an order of magnitude.
Broader Impact Statement and Future Work
This paper aims develop AI-driven, symmctry-aware differential equation simulators to model financial market dynamies, which may also contribute scientific discovery and engineering. This paper also pioneers the realization of Lie symmetries by maintaining the corresponding conservation laws, presenting universal, off-the-shelf solution that is not limited to PINNs or the Black-Scholes equation, but can be extended to a wide range of backbones and differential equations. For future work, we will consider the incorporation of symmetries into network architecture.
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Bo Zhao, lordan Ganev, Robin Walters, Rose Yu, and Nima Dehmamy. Symmetries, flat minima, and the conserved quantities of gradient flow. In The Eleventh International Conference on Learning Representations, 2023.
Liu Ziyin. Symmetry induces structure and constraint of learning. In Proceedings of the 41st International Conference Machine Learning, volume 235, pp. 62847-62866, 2024.
Liu Ziyin, Mingze Wang, Hongchao Li, and Lei Wu. Parameter symmetry and noise equilibrium of stochastic gradient descent. In The Thirty--ighht Annual Conference on Neural Information Processing Systems, 2024.
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A
Appendix
The Appendix is divided into three parts: 1) Section A.1
provides the nocessary definitions and lemmas, 2) Section A.2 includes the general form of the PDE, 3) Section A.3 provides the knowledge about the Lie symmetries of Vašiček, 4) Section 5.3 provieds the some experiments with different Lie symmetry operator conbinations, and 5) Section A.4 presents the theoretical analysis of LSN, including its approximation and gencralization properties.
A.1 Definitions and Technical Lemmas
In this section, we will present the definitions and lemmas required for our subsequent discussions. Definition A.1 (Wiener process). The Wiener process (also known Brownian motion) is continuous-time stochastic process commonly used to model random walks. The standard definition of a. Wiener process includes several key features:
1. Starting Point: The process starts at Wo = 0, indicating that its initial position is zero.
2. Independent Increments: For all 0 S t, the increments W2 W, are mutually independent. This implies that the process memory-less, and its future behavior not influenced by its past.
3. Stationary Increments: For all 0 S s < t, the distribution of the increment W - W, depends only on the time difference - 8, and is independent of the specific values of 8 and t. Mat Mathematically, this is ex- pressed as W,WW, ~ ((0,t-s). where N t-s) denotes a normal distribution with mean and variance t-s.
4. Continuous Paths: The paths of the Wiener process almost surely continuous. This means that the function t + W, is continuous with probability 1.
Definition A.2 (European call options). European call options are financial derivatives granting the holder the right, without obligation, to purchase the underlying asset at predetermined price upon expiration.
Definition A.3 (Lie symmetries (Gazizov & Tbragimov
1998 ) Consider second-order evolutionary PDEs:
Ut - Ft, U(1), u(2) 0,
(13)
... xx), and U(1): U(2) represent the sets where is a function of independent variables and x (.'.... of its first and second-order partial derivatives: u(1) (Uzt, Unn) ((2) (Uatal, Uglz?, -M Maman): Transformations of the variables t, 2, given by:
i= ft,, u. a), z'= x'= ū= h(t,,,u,,a) i= 1....,
(14)
where these transformations depend a continuous parameter These are defined symmetry transfor- mations of Equation [3 if the equation retains its form in the new variables ī, ū. The collection G of all such transformations forms a continuous group, meaning G includes the identity transformation:
i=t, F= T==
ū=
the inverse of any transformation in G, and the composition of any two transformations in G. This symmetry group G is also known as the group admitted by Equation (13). According to the Lie group theory, constructing the symmetry group G is equivalent to determining its infinitesimal transformations:
is t (t, u), / x' + (t, u nd u an xz u).
(15)
For convenience, the infinitesimal transformation Equation 15 can be represented by the operator:
① + (t, x, u) dr + n x, u) X x= (t, 0t
0 0u
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Definition A.4 (Conservation law [haliquu & Motsepa, symmetry
2018 Any Lie point, Lie- Bäckhaa or non-local
0
X 61 u, (1) ,....
+ nn,, u, (1)) ( ....)
dr
Ou
of differential equation
F(z,u,ul) (()) ) =0
(16)
provides conservation law for the system of differential equations comprising Equation 16] and the adjoint equation
DT( =0 0,
o(vF) 8u
F" u, a v0
(0) (())
The conserved vector is given by
0L
ØL
0L
T= = W
Dj
+ D,Dr.
Ouu
Ouij
Öuigh
ØL
ØL
+...] + DWD
ØL
+
Ouis
Frigh
Origh
where W and L are defined as
W Buy,
I= oF u, (.) ,-.,, (())).
Definition A.5 (Relative test error). The relative test error between an approximate solution i(S) and an exact solution (S) on test collocation data S is defined as follows:
Relative test error = ((S) - u (S)
u² u*()
Definition A.6. [Feynman-Kas Kac formula| Del Moral & Del Moral 2004) The Feynman-Kac formula provides critical theoretical framework to establish connection between certain types of PDEs and SDEs. Given a payoff function f (x, t) and defined a discounting function r(æ, t) to calculate the present value of future payolls, u(x, I) a solution to the PDE:
Pu - r(x, t)u = 0, du + u(x, t) Ou + (z,
(17)
0t
dr 2
0x2
with the terminal condition u(a, T') f(x), then the solution u(x, T') this PDE can be represented
E e St" r(Xo.s)ds s(XT) X, = x
(18)
where XT denotes the value of Equation 12 at time T'
Lemma A.7 (Itô's lemma (Hassler 2016 The general form of Itô's Lemma for function (t, Xt) of time and stochastic process X, satisfying a stochastic differential equation is given by
0f 01
0f 1 =
f
Of dW.
df (t,
+
dt
Ox
Ox.2
Fx
Here t represents time, X, is stochastic process satisfying stochastic differential equation dX, putt+ of and 801 denote the partial derivatives of f with respect to
odW, f (t, X2) the function of interest. Br de time and the state variable x, a is the drift coefficient the SDE, is the diffusion coefficient in the SDE and dW, is the differential Wiener process.
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Lemma A.8 (Dynkin's formula. [llebaner 2012 For every Q, let X be the solution to linear PDE Equation ① with affine R - Rd and a Te Igdxd If y E C² (R4) with bounded first portial derivatives, then it holds that (8eu) (x, = (a, t) where u is defined as
R [ (FF)
= =
for forzel, 1E
(19)
where
=. Σ (xx (XX))))) G,,kml
(20)
(Fo) (x(x))(0))
where We is slandard d-dimensional Brownion molion on probabilily space (2, F, P, (F.))E(o.))) and F is the generator of X..
Lemma A.9 ((De Ryck & Mishra 2022 ). Let E 2, let be probability measure on Ω = [0, IJ let f - R(W + +1), R(W + 1)] be a function and let fo Ω -1---,,- E be tanh newral networks with at most L - 1 hidden layers, width at most W and weights and biases bounded by R. For every <<<>, 1, 1, holds for the generalisation and training ervor Equation 12 that,
if N2 64d(L + 3) In
[4Vd+ 4RW
P (EG (0) S (0(5)
A.2 General PDEs
In this section, we will demonstrate the transformation of the BS equation and the Vašiček equation into a general form, i... Equation ①
Black-Scholes equation. As detailed in the main text, the specific expression of the Black-Scholes Equation is
Sur ot, + = 8 + rT, dzz ru, 0, (z,, t.) E Ω [0, Ba; u, x1) max(# xz, K,0), 0),
E Ω t, E [0, T].
(21)
(1,,, 0) = 0,
Let's t= T [T, 0], x, Ω. Then the BS Equation initialbbunndrry value problem [Cervera 2019).
21 be transformed into more generalised
Su + 000222200 + rx Ou कt
-r==00 0,
(x, t) E Ω x [0,2], x E Ω
85
u(0, x) = max (x K, 0),
(22)
0t. = 0,
t E [T.0].
Here L(u] in Equation [1 for Equation 222 is L(ul f0²22 u(x.)) ((x) rz, v(z) = and (p(x) maz( K,
0a(z.1)
ru(x,t) with ((x) ox,
Vašiček Equation (Privault 2022). In a financial market characterised by short-term lending transactions between financial institutions, the evolution of short-term interest rates can be modeled by the following SDE:
dr: >(B xx)dtt odW..
where X, A,B> > 0, a are constants and W, is the Wiener process. Using the Feynman-Kac formula we can obtain the Vašičok pricing equation which is used to price risk-free bonds u(x; t):
Ou + du + A(B - xx) Ou az
+ yzu = 0 Ω x [0. Ω x T.
St az a(I,T)=
(23)
u(x, t) = (x,
aΩ x [0, T].
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where = 2 y= -1 and t)) is the boundary conditions. The zero-coupon bond price in the Vašiček pricing model is given by (Khalique & Motsepa
2018):
u(x, 1) = (((T-())te ()x0((T-))
(24)
where C(t)= (1 > and A(t) - o² -2-2XR0 113
2xa
t++
0² 0²-123 e-
12se
e-Ar-
o²
Similarly, we can express the Vašiček equation in general form as follows:
uz t) = C[u],
for all (a, t)
[0,T],
m(0, x) (
for all E Ω
(25)
u(y,t) (y,
for all (y, t) E 0 [O,T],
where L[u] in Equation 11 for Equation 23 is L[u]
= au2e + A(3 - z)u2 + you with o(x) = 22a,
((x) X(3 sr), vz)) yz and (xx) 1.
A.3 Lie Symmetries of Vašiček Equation
Lie Symmetry Operator. Lie symmetry operator is a major mathematical tool for characterizing the symmotry in PDEs (see Definition A.3 (Paliathanasis et al. (016). The Lic symmotry operators (Khalique
& Molsepa
2018 of Vašiček Equation
23 are given by the vector field
G = 0(1,,,)))
x)
G1
Ou
Øt.
0
e
2h
2e
a
G2 2h1
+
(>2x 2ary -3x2
+
(a²g² + 22. - a 3a7d22+ z)²)
dt
dr
ax²
Du'
G3 =
2h
1
(z -B) -2a7)
θ
yu
(X2x an)
0
0t
Oz
X²
Du
yu 0
G4 A
+
(-an- BX² X²x)
GGg ~
GG
u
dr
ad
Ouj
Ox
Ōu
Ou
(26)
These Lie symmetry operators not only provide deeper insight into the structure of the PDEs but also form the foundation for deriving conservation laws associated with these equations.
Conservation Law. To ascertain the Lie conservation law operators for the Vašiček Equation necessary analyze its adjoint equation, as follows
23 it is
Ov dv - A(x - {3) dv
- a 2x.
Ox (A+ = =.
(27)
Öt
where / # O is a new dependent variable v = et+or with =q² - -Bq + X and
(Here, only one
example presented for illustration purposes, although there cxist numerous solutions to this set of adjoint equation). For illustrative purposes, we choose the relatively simple operator G5 and Go as examples of the Vašiček Equation (26 and provide the corresponding conserved quantities (Khalique & Motsepa, 2018):
Me 1 -My
du
(u, x, t)
yu
71
dr
Ov
Ou
X dv dx
Ou 0t
T:. (u, x. 7x
dr
7u
3Bv)
7v
>
Ox
dr
T (,,,,,,11 =wv
du
{ M(x - -3y + a Ov
To (u, x),)
=a
v u
Dz
0r
In Section 5.3 we validate the general applicability of LSN by extending it the Vašiček model and showing the adaptability of different Lie symmetry operators.
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A.4 Theoretical Analysis
Given the wide range of choices for Lie symmetry operators, use the BS equation with the selected lie a
operator G2 = Bh of Equation (5) as an example to theoretically demonstrate the effectiveness of our
method. The corresponding conservation law Equation
6 is as follows,
RLie [ū] ( (a),
(28)
where
2bù
(a) -yl(t) -+ -- I σ²x
(29)
2r
Th (g) = a,1(1) + al' (1) + g(l) būe rt + 0r +
xe rt
02x
Performing operator calculations with the conserved quantities (I2,1%) substituted into the Equation yields
28
2bû,
tM 2rbů π2x e 02x e e-
(11 = -Qzll(t) (t) +
2rûς
2ru
2r
( = (tt)(t)) + (t)e(t) - bú rt + Üiz
σ²x
0²x²
re
De
+
Gr +
0²x
e
Therefore, we have
2h04 e- büge + brügze
2rbůg
2rhů
( (a)
rt
+
r
rnt + b
d²x
σ²
o²r
2b
2rb
2rb
lote -'xx +
"un
-ra
02x
x²
2be Tt Tt
x²x²a
ana
rrad rū
02x
Since b is arbitrarily chosen, let's set b Imin- Where min represents the smallest x-coordinate among the points in the configuration set. And (z, E f2x x [] represents bounded interior region, where and are within the specified domain Ω and time interval [0, 7] respectively. Therefore, there exists positive
number M > 0 such that
2bc rt
2bc rt 2
mnn 11
2ini (Imine-
2 1
2e 2e- rT
0 <
σ²x r
=
x2xmin
0² = -M.
(30)
Therefore,
Cria((]] [ū]
= =0 = + (
= || 2be TT
+
00
2z2un + rrū, - rū 1|?
22x
2
(31)
I Ml
+
00²
2
+ radd - rāll
2
= [0]]]1
A.4.1 Approximation Error Bounds of LSN
The PDE in Equation (1] is a linear parabolic equation with smooth coelficients, and conclusions about the existence of a unique classical solution to the equation, which is sufficiently regular, be derived using standard parabolic theory. If u is considered classical solution, then the residual concerning should be zero.
Rx] =0, R.(u)(1,t) =, Rx[u[(x) = 0, Riie t) =0, Vr E Ω y E 0..
(32)
Here RLie (z. ae list. several crucial lemmas used to prove the approximation error of T.SN.
ot (m + 2 ugx + rru, - ru) .يΣ R4
[u] (x,t) = (with a2z
" # +.) We first
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Lemma A.10. Let T > 0 and y, y,d, E N with 2 224 Suppose W (00,))x [0, ) is the solution to linear PDE equation 7 hidden layers of width at most O (s-d/(6-2-11) such that & ( S..
Then, for every > 0 there erists a lanh neural network ū Uge with two
Proof. We extend the proof of the Theorem 1 in De Ryck & Mishra (2022 to the LSN algorithm with regularization terms incorporating Lie symmetries. There exists a tanh neural network a with two hidden layers of width at most O (e (e-d/(6-2-1)) such that
[[=3= S..
Due the linearity of PDFs (where Equation 1 is linear equation with respect to u), it immediately follows that
S and [Rrieuulzza(0 ((0. (o.T() I M [R.[u]]ra(o ()xx)))) s. By employing a standard trace inqquality one can establish similar bounds for R&(u] and Ry[u]. Consequently, it directly follows that & (0)) S.
]
This lemma shows that the structure risk of LSN in Equation 12 can converge to zero. To address the challenge of the curse of dimensionality in structure risk of LSN Equation I22 bounds, we will leverage
Dynkin's Lemma A.8 which establishes a connection between the linear partial differential Equation [I and the Ttô diffusion stochastic equation. Next, we will extend the proof for PINNs from De Ryck & Mishra (2022) to LSNs to demonstrate that the loss for LSNs can be made infinitesimally small.
Lemma A.11. Let a. B, w. W.T.T > O, and p > 2. For any E N, define Da [0, 1]d and consider Pa C (I2) with bounded first partial derivatives. Given the probability space (Sa x [,, p), and let ua E C2.1 (Sa [)) be a function satisfying
(Dtua) (x,, 1) = L [ua] (x. (), ma(x, 0) = pd(()) Lise [ual (xx, =0 =0 for all (r, 1) E Qa x [0.
Assume for every E, 6,6 0, there ewist hyperbolic langent (lank) neural nelworks such that
1|ea - -e-GGallccus.. SE and - (a(l-ao0) (ox S.
(33)
Under these conditions, there exist constants C, 0 such that for every E 8> O and d E N, constant > 0 (independent of e ) and a tanh neural network V Ve,d with at most C (dpa)) e max (b) neurons and weights that grow at most C (dpa)) e- max[{.8p+6] for E + 0 can be found such that
Ve L ( (P + Meed - uall uall (Paxx
(34)
| uall (0ll:1000)0000011 + Crie (M (lll S
where pa is defined as
| Xf (f |-
Pd = max sup ZEft (0.7)
<0.
(35)
e<t
In this context, X denotes the solution, following the Itô interpretation, of the stochastic differential equation (SDE) specified by Equation equation is defined as follows: Given measure space (S,F, where q > O, for any F ) function f: 1.0++RR Rdd
13 Here q > 2 remains independent of and the norm || -llo II (P.1-)laa))
[/ [I[[(W)] [I((W))(
(36)
Proof. The main proof follows directly from Theorem 2 in De Ryck & Mishra 2022 where
|Las ( We.allleeaaaxx... S M V
a. a L [Ve.d Weadll S..
(37)
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According to Remark 2 by De Ryck & Mishra 2022) it is indicated that the assumption conditions in the Lemma A.11 are easily satisfied after modifications for the BS equation.
Theorem A.12. Let u be classical solution to linear PDE described in Equalion 1 with p E C (2; R4)
and a E C2 (9; Rdxd), let M -2a0² u CCC x (0,TRR) and define the residuals according Equa-
tion 11
Then,
1|u u " S0I [0 [Raa[[]]l22 ||R
(38)
+C2 RR, (1) 11810(80000))))) + Css [|, (|
where
Co 2 Σ
aus (oo") 1=0000000000
1,5=1
C = Tel Te( div +1+ -1lu0l==
(39)
C2 = 2 1100 @o Vzlu v)), (i)):))))
C3 = 2||/|loc +1 (1
M)
Σ iik-=
||a (ou0)))) (0000))lll=x000000000
Proof. Let à = Integrating (t, over Ω and rearranging terms gives
dt / 1i/ 2/2 / Trace (o² (0]) ūdr / p [ulzddidd + / v/0ffdx R, R.(iddde,
d
(40)
where all integrals are understood as integrals with respect to the Lebesgue measure on Ω and an, and where Ju represents the Jacobian matrix, which the transpose of the gradient with respect to the spatial coordinates. Following the derivation by Theorem 4 of Ryck & Mishra (2022), we can similarly show that for the first term
JA Trace (ao" H₂ [i)) ådr E Jan (oo" Jn()))), a (e: A))de. Ja[ūjo (J](ū)o)* de + Jan |R|ul] da + 3 Sn id,x,
(oo"HH[(0]
(41)
for the second term
I JrJz [1] (iddd v
div pallo ( úidxx +
1 (rdx+211111 21111110
/ [R,[u]] da,
(42)
2
jon
for the fourth term
/ R4 udx
1 / R. R.(ūū]2 dx
1 S id
<
(43)
n
2 n2
where â denotes the unit normal on 02. 15 i.j, h < d and
C Σd d
- (00 J_ a1 (7))
C9 = Σ Eij
||a, (axx 5
(00x00
(44)
C3 = -
Ois (oo?) -xx00)))
As for the third term of Equation we obtain
/ u/il2ds S 110/1 / P
(45)
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Integrating Equation (40) over the interval [0 C [0, T], using all the previous inequalities together with Hölder's s inequality, we find that
/ |Rolu|| dr + c1 (1 RR. [R] doet
1/2
1 [1((z,))) d) -
+
R
drdt
(46)
/ / 10.0
/
/ |ā(x,, P 10.
+ (c2 11allso)
[R[vv]] dodl + (e3 + div ulloo + +1 (1llox)
Referring to Equation
" (ut + fo + rour - ru) with 40 we can transform operator Rise [ū] = 0xx 0²x
*0, i.e., R.( uu + 7 o Uaz raug ru 2be 0x
2be 2be-rt Reio [t]] into the following form,
d
/ |a dx
2 / Trace (o'Hz(a)) ūdx + S pJzlāj + ulālda
1
/
o²x
2be Rese
dt 2
(47)
l// Trace (o² ude + S. pd [djl(ajada + vla dx wlaldd
/ RLin[ū]d..
$ 2
+
M
The proof is ultimately established by using Using Grönwall's inequality and integrating over [0, T).
Remark A.13. Theorem A.12 states that by optimizing structure risk the network's output can approximate the exact solution, while Lemma A. confirms that structure risk can be minimized. This verifies the numerical approximation the LSN to the exact solution.
A.4.2 Generalisation Error Bounds of LSN
We set general configuration let Ω C Rd be compact and let Ω - R, us : + IR be functions for all 0 E O We consider the exact value of the PDE equation 1 and uo the approximation generated by LSN with weights 0.
Let N E be the training set size and let S = (21,.... mM} E Ω be the training set, where cach 2; is independently drawn according to some probability measure j on Ω. We define the structure risk and empirical loss as
N N
((0) ( [1o(2) - u(2)) dp(z), L(,
uuo (2+) 2 (2,)1,, (*(S) mig
min L(,, S). Age
(48)
Lemma A.14. Let d, EN with R21, 1, and define M = -(20)=)) Consider us [0, 1]2 R, where BEO representing tanh neural networks with at most L --1 hidden layers, each with width of at most W, and weights and biases bounded by R. Let L and La denole the structure risk and empirical error,
0²
respectively, for linear general PDFs as in Equation 1 Assume max {llyploc. [[ I maxgee luolloo
Denote by 2 the Lipschitz constant of L for q == i,,,8. 8. Then, it follows that
L 2 25+2 2 1 W6L-6
+ Σ (o(=))
where C' (1 M) max 1 +
iJ=1
Proof. Similar to Lemma 16 in De Ryck & Mishra
2022 we have the following:
RR [uel (t, - R₂ [ (t, x) s [u(=)] luo - wh + (1+ lm(z)))) = +100))0(0)) H - H
4a (1 + |o(x))) + [m(x)] + +o0b" = - xo
And have
Rute [ue] (1,x) Ruso [0]] (1,z) < M RR [uo] (1,x:) R, [Φ0] (1,
(49)
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where we let 1. denote the vector p-norm of the vectored version of a general tensor. Next, set e 0 and max (lle)l>>, [|vb]loc] I max
| for
1,8 8 that
max R [ee] II (d
max
| RRie [ || 30 S4aC, (d 7,2 33l-3
(50)
max
R [ |l
22WR, R,
where C = max (1 + + (o()) Combining all the previous results yields the bound.
We can then obtain the generalization bound of LSN as follows.
Theorem A.15. Let L, W E,, R2 1, L, W 2 2, a, b E R with a < b and let us 10,1 be tanh neural networks with at most L 1 hidden layers, width at most W, and weights and biases bounded by R. For q i,,,8, let La and La denote the LSN structure risk and training error, respectively, for linear general PDEs as in Equation 11 Let Gg be such that (0)0,)), (*))) E [0, cyl, for all 0 O and S Assume max 1l [bol max /olloo and define the constants
2
C= (1+++M)
max
+
Σ 1=1
(o))d +
Σ 1=1
+
Σ (o(0)
zef
1,1=1
Then, for any e 0, holds that
2ddL2 22
C(d + 7)
ca set C
if,
In Rc V
(51)
c²
Proof. The proof follows the generalization analysis of PINNs (De Ryck & Mishra
2022). Setting
2
C (1 + M) max 1 Σ
[x))+++ + Σ [m(x)++ Σ (0(0)o(1))))))
(52)
ZED
1=1
1=1
Lemma A.14 with + R. t c..2 25 + 7) 886- 6 and k 2dLW2 We then arrive at
(Lemma A.9)
kln
4a -
In
26g 2
< 6kLln
RC // //a447
12dL2 In (Ac, (Ac,Ra RW V [(d47)
29


Reviews:
[{"review_id": "SU3vcZDZvA", "reviewer": "Reviewer_QiWW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**\n- The paper is well-organized and well-written, making it easy to follow.\n- LSN demonstrates a significant performance improvement over existing methods.\n\n**Weakness**\n\nMany prior works have incorporated additional residual objectives into PINN to enforce specific constraints. These works significantly improves the performance for the task in the specific domain. I was convinced that LSN effectively enhances PINN for the Black-Scholes (BS) equation and achieves superior performance. However, I believe some aspects require further verification:\n\n- **Comparative Analysis for LPS** The comparison with LPS is currently limited to settings where $\\sigma\\approx 0.5$ and $r \\approx 0.1$. A more comprehensive evaluation across a wider range of parameter values would strengthen the analysis. Specifically, could authors provide results for PINN variants, LPS, and LSN across $\\sigma=0.05, 0.1, 0.2, 0.5$, $r=0.01, 0.02, 0.05, 0.1, 0.2$ (total 20 experiments for each methods)?  These values are particularly relevant given that, in real-world scenarios ($r\\approx 0.2$. $\\sigma=0.2$).\n\n- Moreover, I believe LPS should be trained with various $\\lambda_4$ for the fair comparison. I suggest using $\\lambda_4 = 1e-3, 1e-4, 1e-5$.\n\n- The number of training iterations varies across experiments in the BS setting. Would it be possible to standardize this to 200K iterations for all methods to ensure comparability?\n\n- **Interpretability of LSN's advantage**: Both LPS and LSN incorporate residual loss terms related to Lie symmetry. Could authors provide an intuitive or practical explanation, possibly supported by toy experiments, to illustrate why LSN performs better?\n\n- **Additional Experiments on Different Dataset**: Could authors include experiments on European and American option pricing following [1]? This would further demonstrate the applicability of LSN beyond the Black-Scholes equation.\n\n[1] Ashish Dhiman, Yibei Hu, Physics Informed Neural Network for Option Pricing, 2023.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes the Lie Symmetry Network (LSN), a Physics-Informed Neural Network (PINN) designed to characterize and preserve the Lie symmetries of differential equations. LSN enhances the standard PINN loss by incorporating an L2 error term derived from Lie-symmetry-based conservation laws (or Lie symmetry risks), ensuring that learned solutions adhere to fundamental invariance properties. Empirical results demonstrate that LSN achieves state-of-the-art performance on the Black-Scholes equation and consistently outperforms Lie Point Symmetry (LPS) methods across various settings.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Summarization of the requests described in Weakness section:\n\n- PINN, LPS, LSN experiments on $\\sigma=0.05, 0.2, 0.5, 0.8$, $r=0.02, 0.05, 0.1, 0.2$ with $\\lambda_4 = 1e-3, 1e-4, 1e-5$, Total iterations of 200K.\n\n- Could authors provide the reason why LSN performs better than LPS?\n\n- Could authors include experiments on European and American option pricing?", "broader_impact_concerns": ".", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "aOSWDc3gna", "reviewer": "Reviewer_eWZt", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n* The proposed solution for incorporating Lie symmetries of PDEs is simple and straightforward, though deriving the exact form of the Lie symmetry risk requires some prior knowledge.\n* There are detailed descriptions of the experiment setup, and the results look promising for the specific case of modeling the price of a call option.\n\nWeakness:\n* The application of the proposed method is restricted to a single PDE system.\n* Despite the extensive experiments, there is no intuitive explanation of why the proposed method could work better than LPS (Akhound-Sadegh et al., 2024), since both methods use loss terms based on Lie symmetries, only the exact forms of the loss terms differ.\n* The derivation of the conservation law is not included in the main text. I think this is an important step of the method, and it would be better if some key points of the relevant discussion in Appendix A.3-5 could be moved to the main text.\n* There are some missing references about exploiting Lie point symmetries for learning in differential equation systems.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a new loss term, i.e. Lie symmetry risk, based on conservation laws derived from Lie point symmetries of PDEs for PINNs. The Lie symmetry risk is added to standard PINN losses to ensure the neural PDE solver learns solutions that respect the underlying symmetry of the PDE system. Compared to previous works incorporating Lie symmetry into PINN learning by optimizing the deviation from Lie's infinitesimal criterion (Akhound-Sadegh et al., 2024), the Lie symmetry risk used in this paper seems more indirect, since it requires the additional step of deriving the conservation laws from the Lie symmetries. The method is applied to the specific scenario of modeling the price of a call option in terms of the price of the underlying asset and time. Experimental evidence shows that the proposed loss term results in better prediction of PDE solutions evaluated on the test data points.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "* Add some intuitive explanation in the experiment section on why the proposed method works better than LPS (Akhound-Sadegh et al., 2024).\n* In Figure 2, change the label on the Y-axis to \"t\".\n* To demonstrate the overall effectiveness of all PINN-based methods, can you also include an error measure and an error plot for a dummy baseline, e.g. a second-order approximation of the option price using the Greeks?\n* Add more details in Section 4.1 on how the conservation law corresponding to $G_2$ is derived, and on the reason why the specific one-parameter subgroup $G_2$ is chosen.\n* Section 4.2 can be made more concise since it only involves the standard procedure of creating the empirical risk for optimization.\n* Consider including more references in the Lie symmetries subsection. The authors may also find the following papers relevant as they all aim to incorporate Lie symmetries in learning for differential equation systems.\n    * Brandstetter, J., Welling, M., and Worrall, D. E. Lie point symmetry data augmentation for neural pde solvers. In International Conference on Machine Learning, pp. 2241–2256. PMLR, 2022.\n    * Otto, S. E., Zolman, N., Kutz, J. N., and Brunton, S. L. A unified framework to enforce, discover, and promote symmetry in machine learning, 2023.\n    * Dalton, D., Husmeier, D., and Gao, H. Physics and lie symmetry informed gaussian processes. In Forty-first International Conference on Machine Learning, 2024.\n    * Yang, J., Rao, W., Dehmamy, N., Walters, R., and Yu, R. Symmetry-informed governing equation discovery. In Advances in Neural Information Processing Systems (NeurIPS), 2024.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "crPeRtVAG9", "reviewer": "Reviewer_KZfX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- The approach is relevant and the idea is sound.\n- The paper structure is fine and I was mostly able to follow the paper.\n\nWeaknesses\n- The paper introduces a new method that should work for somewhat general PDEs with Lie symmetries. I find it disheartening to just test this method on a single PDE. This is a major drawback in this paper. (At least, it is made very clear, beginning from the abstract, that the paper takes this restriction.)\n- There is no comparison \n  - against neural operator approaches and \n  - against state of the art approaches from numerical analysis.\n (PINN-based approaches have other advantages, perhaps you should stress them)\n- No mention of computation times is given. Is the new approach at least comparable in computation time to previous approaches?\n- I get that this is not a paper going deeper into SDEs. However, one should at least mention that applying the Feynman-Kac formula (e.g. form (3) to (4)) looses information. The deterministic PDE in (4) is just an expectation of the SDE in (3). This might seem like a minor point, but the paper claims (e.g. in the first line of the conclusion) that such SDE are being solved and not their expectations computed.\n- The paper gives many citation (which is obviously a good idea), but is usually unprecise in its citation. For example the citation into (Rotman, 2012) on p5 cites a long book, and should better cite a specific chapter/page/definition. Or (Akhound-Sadegh et al., 2024) has the wrong year, see https://proceedings.neurips.cc/paper_files/paper/2023/hash/8493c860bec41705f7743d5764301b94-Abstract-Conference.html.\n- While you cite state of the art methods, you do not describe differences. In particular, (Akhound-Sadegh et al., 2024) look very similar to your approach. There, I would like to see the differences be clearly described.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper suggests an approach to include Lie symmetries into PINN loss functions for dealing with particual differential equations. The approaches are tested on SDEs from financial areas, which are transformed into PDEs via the Feynman-Kac formula. Superiority to some state of the art methods are shown on the Black-Scholes equation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Obviously, I would expect that all weaknesses are addressed in some form.\n\n- p 2 l 6: I find it hard to follow to give the Lie symmetry without stating the Black-Scholes equation or the Lie group.\n- p 2 l 7: \"D_\\cdot\" might be mistaken for \"D.\". (See also Definition 4.1)\n- p 4 l 4: \"of partial\" -> \"of a partial\"\n- (1): This is not a general PDE. This is a \"univariate evolution PDE with initial conditions and Dirichlet boundary conditions\".\n- p 5 l 5: \"when groups are also differentiable manifolds, they are referred to as Lie groups\" is wrong\n- p 5 line after (5): the comma in \"Z = r − σ2/2, ϕ(t, x)\" makes it hard to parse this as two symbols.\n- p 7: While it is pretty clear form the context of PINNs, one might add a (short) explanation how $\\mathcal{R}_{Lie}$ is calculated\n- p 7 l -9: What does \"Gaussian sampling\" mean here?\n- The capitalization of the titles in the bibliography is off. While I grant that lower cases might be used, names should still be capitalized.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Overall, the reviewers have appreciated the work. Their concerns were mostly resolved during the feedback phase, and all of them have acknowledged the same. However, a few minor concerns still remain, which should be addressed in the paper.

1. It might still be helpful to add an experiment for a third PDE for the camera-ready version. A reviewer has suggested this.

2. A reviewer raises critical concerns about the motivation behind using the conservation loss instead of the established Lie symmetry loss term. It might be a good idea to explain this better in the paper. Some additional experiments in this regard would be useful. I am not sure how feasible this might be, but do take a stab at them.

Based on the reviews and discussions, I believe the paper is suitable for publication in TMLR with minor revisions.
Decision: Accept (poster)