Gradient Networks for Universal Magnetic Modeling of Synchronous Machines

1 Gradient N etw orks f or U niv ersal Magnetic Modeling of Sync hronous Mac hines Jun yi Li, Tim Foißner , Floran Mar tin, Antti Piippo, and Marko Hinkkanen, F ellow , IEEE Abstract —This paper presents a phy sics-informed neural net- w ork approach for dynamic modeling of saturable synchronous machines, including cases with spatial harmonics. W e introduce an arc hitecture that incorporates gradient netw orks dir ectly into the fundamental mac hine equations, enabling accurate modeling of the nonlinear and coupled electromagnetic constitutiv e rela- tionship. By learning the gradient of the magnetic field energy , the model inherently satisfies energy balance (reciprocity conditions). The proposed architecture can univ ersally appro ximate any ph ys- ically feasible magnetic beha vior and offers sev eral adv antages o v er lookup tables and standard mac hine learning models: it requir es less training dat a, ensures monotonicity and reliable extrapolation, and produces smooth outputs. These properties further enable robust model inv ersion and optimal trajectory generation, often needed in control applications. W e validate the proposed approach using measured and finite-element method (FEM) datasets from a 5.6-kW permanent-magnet (PM) syn- chr onous reluctance machine. R esults demonstrate accurate and ph ysicall y consistent models, ev en with limited training data. Index T erms —Dynamic modeling, electric drives, gradient netw orks, Hamiltonian, magnetic saturation, neural networ ks, ph ysics-inf ormed machine learning, synchr onous machines. I. Introduction D YN AMIC models of electric machines are essential f or control, estimation, monitoring, and for the design and optimization of dr iv es. The most challenging aspect of machine modeling is the magnetic model, whic h descr ibes the relationship between flux linkages, currents, rotor angle, and electromagnetic torque, assuming magnetostatic conditions [1], [2]. In moder n pow er -dense electr ic machines, magnetic saturation effects are significant and must be incorporated into the models. High-fidelity models that include spatial harmonics are also needed, e.g., in time-domain simulations during the design and optimization stag e. The nonlinear magnetics are modeled using analytical func- tions [3]–[7], lookup tables [8]–[13], or neural networks [14]– [17]. These models can be characterized based on finite- element method (FEM) data [18], laborator y measurements This work has been submitted to the IEEE f or possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. This work was supported in par t by the ABB Oy , in par t by the Aalto Univ ersity House of AI, and in par t b y the Research Council of Finland Centre of Excellence in High-Speed Electromechanical Energy Conv ersion Sys tems. The authors ackno w ledge the use of EPE infrastructure of Aalto School of Electrical Engineering. Junyi Li, Tim Foißner , Floran Martin, and Marko Hinkkanen are with the Depar tment of Electr ical Engineering and Automation, Aalto Univ er - sity , 02150 Espoo, Finland (e-mail: junyi.li@aalto.fi; tim.f oissner@aalto.fi; floran.martin@aalto.fi; marko.hinkkanen@aalto.fi). Antti Piippo is with ABB Oy , Dr iv es, 00380 Helsinki, Finland (e-mail: antti.piippo@fi.abb.com). [19], or automatic identification routines [20]. The accuracy of the anal ytical models is limited, and they are difficult to extend to higher dimensions (e.g., f or spatial har monics or multi-phase machines). Look up tables w ork well in tw o dimensions but suffer from the curse of dimensionality , high memory requirements, and non-smooth output when using lin- ear inter polation. Black -bo x neural netw orks can appro ximate high-dimensional complex maps and require less memor y than lookup tables, but their training still demands larg e datasets, their e xtrapolation capability is limited, and energy balance is not guaranteed. T o address the limitations of blac k -box neural netw orks, ph ysics-inf or med neural netw orks combine data with known ph ysical principles [21]–[24]. Hamiltonian neural netw orks [23], [24] are par ticularl y relev ant in this context, as electr ic machines are por t-Hamiltonian systems [25] with magnetic field energy ser ving as the Hamiltonian. According to fun- damental phy sical pr inciples [1], [2], the cur rent v ector and electromagnetic torque are the gradients of the field energy with respect to the flux-linkage vector and rotor angle, respec- tiv ely . The Hamiltonian neural netw ork arc hitectures [23], [24] model the Hamiltonian as a neural network and obtain the gra- dients via numerical differentiation. This approach improv es data efficiency and ph ysical consistency compared to black - bo x netw orks. Ho w e v er , it s till f aces challeng es related to numerical differentiation of the scalar neural netw ork to obtain gradients [26]. Recent gradient networks [26] directly model conservativ e v ector fields, enabling universal appro ximation of any gradient field without the need for numer ical differentiation of a scalar neural netw ork. This approach allo ws the model to inherently satisfy phy sical law s such as energy balance and reciprocity , while also impro ving data efficiency and generalization. In this paper , we propose a phy sics-inf ormed magnetic modeling framew ork f or synchronous machines that combines fundamental electromechanical dynamics [1], [2] with gradient netw orks [26]. The stator cur rent and electromagnetic torque are directl y modeled as gradients of a scalar field energy function, which guarantees ph ysical consistency by design. W e employ monotone g radient networks to ensure the field energy is con v e x. This proper ty enf orces a unique, inv ertible relationship betw een flux linkages and cur rents, enabling the f ormulation of both current and flux-linkage (dual) maps. A dditionally , F our ier features [27] are incorporated to capture spatial harmonics while preser ving the lossless (conservativ e) field structure. The resulting model offers a universal ap- pro ximation of comple x magnetic beha vior , including cross- 2 Lossless magnetic sy s tem 𝑊 s ( 𝛙 s s , 𝜗 m ) 𝛙 s s 𝑅 s 1 𝑠 u s s 𝜔 m 𝜏 m i s s 1 𝑠 𝜗 m (a) Lossless magnetic sy s tem 𝑊 ( 𝛙 s , 𝜗 m ) 𝛙 s 𝑅 s u s 𝜔 m 𝜗 m 𝜏 m i s J 1 𝑠 1 𝑠 (b) Fig. 1. Electromechanical dynamics of a generic synchronous machine: (a) stator coordinates; (b) rotor coordinates. The blocks 1 / 𝑠 denote integ ration in time. saturation and angle dependency . W e also compare activation functions and propose a computationally efficient 𝑝 -norm gradient activation as an alter nativ e to the commonly used softmax. This paper is organized as f ollo ws. Section II revie w s the ph ysics-based machine model. Section III details the proposed gradient netw ork architecture. In Section IV, the model is validated using measured and FEM datasets from a 5.6- k W per manent-magnet (PM) synchronous reluctance machine. Results demonstrate accurate and ph ysicall y consis tent models, ev en with limited training data. Section V concludes the paper . II. Physics-B ased Machine Model Scalars are denoted b y italic letters (e.g., 𝑥 ), column vectors b y bold low ercase letters (e.g., x ), and matr ices by bold uppercase letters (e.g., A ). All quantities are in per -unit unless otherwise specified. A. Stator Coordinat es Fig. 1(a) show s the block diagram of a generic synchronous machine model in stator ( α β ) coordinates. The corresponding state equations are d 𝛙 s s d 𝑡 = u s s − 𝑅 s i s s (1a) d 𝜗 m d 𝑡 = 𝜔 m (1b) where 𝛙 s s = [ 𝜓 α , 𝜓 β ] ⊤ is the stator flux-linkage v ector , i s s = [ 𝑖 α , 𝑖 β ] ⊤ is the s tator current vector , u s s = [ 𝑢 α , 𝑢 β ] ⊤ is the s tator v oltage v ector , and 𝑅 s is the stator resistance. Fur thermore, 𝜗 m is the electr ical angle of the rotor d-axis with respect to the stator coordinates, and 𝜔 m is the electrical angular speed of the rotor . Assuming a lossless (conser v ativ e) magnetic field sys tem, the magnetic behavior is full y descr ibed by its field ener gy function 𝑊 s ( 𝛙 s s , 𝜗 m ) . Hence, the stator current and electromagnetic torque are giv en b y [1], [2] i s s =  𝜕𝑊 s ( 𝛙 s s , 𝜗 m ) 𝜕 𝛙 s s  ⊤ (2a) 𝜏 m = − 𝜕𝑊 s ( 𝛙 s s , 𝜗 m ) 𝜕 𝜗 m (2b) where per -unit v alues are used. The minus sign appears in the torque e xpression since the positiv e mechanical pow er is defined out of the machine. It can be realized that the cur rent and the electromagnetic torque (with negativ e sign) constitute the gradient of the field energy function. B. Rotor Coor dinates The machine model is more con v enient to e xpress in rotor (dq) coordinates, which rotates with the rotor d-axis. Coordinate transf ormation to rotor coordinates can be e xpressed using the matrix exponential as 𝛙 s = e − 𝜗 m J 𝛙 s s (3) where J = [ 0 − 1 1 0 ] is the or thogonal rotation matr ix. The flux- linkage v ector 𝛙 s = [ 𝜓 d , 𝜓 q ] ⊤ is used as an example, but other v ectors can be transf ormed similarl y . 1) Model Structur e: Using (3), the state equations (1) can be transf ormed to rotor coordinates as d 𝛙 s d 𝑡 = u s − 𝑅 s i s − 𝜔 m J 𝛙 s (4a) d 𝜗 m d 𝑡 = 𝜔 m (4b) where i s = [ 𝑖 d , 𝑖 q ] ⊤ is the current v ector and u s = [ 𝑢 d , 𝑢 q ] ⊤ is the v oltag e v ector . Fig. 1(b) sho w s the cor responding block diagram. The field energy function can be expressed in rotor coor - dinates, 𝑊 ( 𝛙 s , 𝜗 m ) = 𝑊 s ( 𝛙 s s , 𝜗 m ) . Applying the coordinate transf ormation and the chain rule (see Appendix A), the magnetic model (2) becomes i s =  𝜕𝑊 ( 𝛙 s , 𝜗 m ) 𝜕 𝛙 s  ⊤ (5a) 𝜏 m = i ⊤ s J 𝛙 s − 𝜕𝑊 ( 𝛙 s , 𝜗 m ) 𝜕 𝜗 m (5b) For a conservativ e magnetic system, the current map i s ( 𝛙 s , 𝜗 m ) is monotone in 𝛙 s [1]. Mathematicall y , this means that the incremental inv erse inductance matr ix 𝚪 s = 𝜕 i s / 𝜕 𝛙 s is positiv e definite, cor responding to a strictly conv e x energy func- tion 𝑊 ( 𝛙 s , 𝜗 m ) with respect to 𝛙 s . In contrast, the dependence on 𝜗 m is g enerall y non-monotone and periodic due to rotational symmetry . 2) q-Axis Symmetry Without Spatial Harmonics: If spatial harmonics are omitted and the d-axis is aligned with the PM flux, the field energy is ev en with respect to the q-axis flux linkage, i.e., 𝑊 ( 𝛙 s ) = 𝑊 ( C 𝛙 s ) C = diag ( 1 , − 1 ) (6) where the matr ix C conjugates the q-axis flux linkage. T o enf orce this symmetry , w e define the symmetrized energy function 𝑊 ( 𝛙 s ) = 1 2  𝑊 ( 𝛙 s ) + 𝑊 ( C 𝛙 s )  (7) 3 a ⊤ 1 𝛔 ( · ) a 1 𝑏 1 a ⊤ 𝑛 a 𝑛 𝑏 𝑛 A 0 b 0 x g ( x ) a ⊤ 𝑁 a 𝑁 𝑏 𝑁 𝑧 1 𝑧 𝑛 𝑧 𝑁 𝜎 1 𝜎 𝑛 𝜎 𝑁 Fig. 2. Gradient netw ork used for the magnetic models. f or which the condition (6) holds by construction. The sym- metrized energy preserv es both the conser v ativ e structure and con v e xity , since reflecting and av eraging conv e x functions yields a con v e x function. Omitting spatial har monics and using the symmetr ized energy (7), the magnetic model (5) simplifies to 1 i s = " 𝜕𝑊 ( 𝛙 s ) 𝜕 𝛙 s # ⊤ (8a) 𝜏 m = i ⊤ s J 𝛙 s (8b) This model guarantees the reflectional symmetr y e xpected in the absence of spatial har monics. The cur rent map satisfies 𝑖 d ( 𝜓 d , − 𝜓 q ) = 𝑖 d ( 𝜓 d , 𝜓 q ) and 𝑖 q ( 𝜓 d , − 𝜓 q ) = − 𝑖 q ( 𝜓 d , 𝜓 q ) . Consequentl y , 𝑖 q ( 𝜓 d , 0 ) = 0 holds f or all 𝜓 d . When spatial har monics are included, the field energy depends on the rotor angle and the cur rent and torque become periodic in 𝜗 m . The strict q-axis symmetry need not hold in that case. 3) Co-Energy-Based Dual Model: Alternativel y , the dual model 𝛙 s ( i s , 𝜗 m ) can be f or mulated. The field ener gy 𝑊 and co-energy 𝑊 ′ are related through the Legendre transf orm 𝑊 ( 𝛙 s , 𝜗 m ) + 𝑊 ′ ( i s , 𝜗 m ) = i ⊤ s 𝛙 s . This relationship yields the dual model 𝛙 s =  𝜕𝑊 ′ ( i s , 𝜗 m ) 𝜕 i s  ⊤ (9a) 𝜏 m = i ⊤ s J 𝛙 s + 𝜕𝑊 ′ ( i s , 𝜗 m ) 𝜕 𝜗 m (9b) Note that the sign of the par tial der iv ativ e ter m in the torque e xpression is positiv e. The choice between the energy-based model (5) and the co- energy -based dual model (9) depends on the application, i.e., whether the cur rent map i s ( 𝛙 s , 𝜗 m ) or the flux-linkage map 𝛙 s ( i s , 𝜗 m ) is pref er red. Due to strict conv e xity , these maps are in v er tible. 1 Naturall y , this model includes the magnetically linear case, where the field energy is 𝑊 ( 𝛙 s ) = ( 𝛙 s − 𝛙 f ) ⊤ 𝚪 s ( 𝛙 s − 𝛙 f ) / 2, with the constant inv erse inductance matr ix 𝚪 s = diag ( 1 / 𝐿 d , 1 / 𝐿 q ) and the PM-flux vector 𝛙 f = [ 𝜓 f , 0 ] ⊤ . In this case, the model reduces to the conv entional linear relationship i s = 𝚪 s ( 𝛙 s − 𝛙 f ) . 0 1 2 𝜎 𝛽 = 0 . 01 𝛽 = 0 . 1 𝛽 = 1 − 2 − 1 0 1 2 𝑧 0 . 0 0 . 5 1 . 0 d 𝜎 / d 𝑧 Fig. 3. Elementwise squareplus activation 𝜎 in (11) and its derivativ e d 𝜎 / d 𝑥 at different values of parameter 𝛽 . The shape of the algebraic sigmoid (12) is the same as the derivativ e of the squareplus, but shifted v ertically and scaled. III. Proposed Magnetic Models Based on Gradient Networks A. Gradient Ne twor ks 1) Structure: Gradient networks can univ ersally appro xi- mate an y monotone conservativ e field [26]. W e use an arc hitec- ture with 𝑁 hidden units, illustrated in Fig. 2. It can be e xpressed as z = Ax + b (10a) g ( x ) = A 0 x + b 0 + A ⊤ 𝛔 ( z ) (10b) where z = [ 𝑧 1 . . . 𝑧 𝑁 ] ⊤ is the pre-activation v ector , A ⊤ = [ a 1 . . . a 𝑁 ] is the transposed w eight matrix, b = [ 𝑏 1 . . . 𝑏 𝑁 ] ⊤ is the bias v ector , and 𝛔 ( z ) is the activ ation v ector . The linear output ter m is defined by the bias v ector b 0 and the symmetric positiv e semidefinite matrix A 0 . The activ ation v ector is selected such that its Jacobian J 𝛔 = 𝜕 𝛔 / 𝜕 z is symmetric and positiv e semidefinite. If elementwise activa- tions 𝛔 ( z ) = [ 𝜎 1 ( 𝑧 1 ) . . . 𝜎 𝑁 ( 𝑧 𝑁 ) ] ⊤ are used, they are non- decreasing. The netw ork (10) is inherentl y conser v ativ e, i.e., it has symmetric Jacobian J g = 𝜕 g / 𝜕 x , see Appendix B. Further more, it is monotone since the Jacobian J g is positive semidefinite. Consequentl y , there e xists a con v e x state function 𝑊 ( x ) such that g ( x ) = [ 𝜕 𝑊 ( x ) / 𝜕 x ] ⊤ . How e v er , the state function 𝑊 ( x ) does not need to be explicitl y modeled, whic h is a ke y advantag e of the gradient network arc hitecture. 2) Elementwise Activ ations: The activ ation function 𝛔 ( z ) can be chosen in v ar ious w a ys. The simples t choice is element- wise activations, where each component of the output depends only on the corresponding component of the input. T ypicall y , the y are computationally more efficient than vector activ ations. Ho w ev er , elementwise activations cannot universall y appro xi- mate all monotone conser vativ e fields. They represent only a subset of such fields, cor responding to gradients of sums of con v e x r idg e functions [26]. Elementwise activ ations must be selected to matc h the saturation characteristics of the targ et map. Hence, different 4 − 2 0 2 𝑧 1 − 2 0 2 𝑧 2 0 . 0 0 . 5 1 . 0 𝜎 1 (a) − 2 0 2 𝑧 1 − 2 0 2 𝑧 2 − 1 0 1 𝜎 1 (b) Fig. 4. V ector activation 𝜎 1 ( 𝑧 1 , 𝑧 2 ) visualized in two-dimensional case: (a) softmax (13) with 𝛽 = 1; (b) 𝑝 -norm g radient (14) with 𝑝 = 4 and 𝛽 = 1. activation types are used f or the cur rent and flux-linkag e maps. For simplicity , we employ the same activ ation function for all hidden units within each netw ork. For modeling cur rent maps, rectifier -type activ ations, such as softplus or alg ebraic squareplus, can be used. The squareplus is giv en b y [28] 𝜎 ( 𝑧 ) = 1 2  𝑧 +  𝑧 2 + 𝛽  (11) where the positiv e parameter 𝛽 affects the shape around zero. Fig. 3 show s the shape of squareplus 𝜎 ( 𝑧 ) and its derivativ e d 𝜎 / d 𝑧 = ( 1 + 𝑧 /  𝑧 2 + 𝛽 ) / 2 at different values of 𝛽 . The derivativ e smoothl y transitions from zero to one, resembling the saturation characteristics of magnetic materials. The sq uareplus function is computationally more efficient than softplus. It is also inherently numericall y stable f or lar ge in puts. For modeling flux-linkage maps, sigmoid-type activations (such as tanh) are more suitable. W e use the algebraic sigmoid activation 𝜎 ( 𝑧 ) = 𝑧  𝑧 2 + 𝛽 (12) where the positiv e parameter 𝛽 affects the slope around zero. The shape of (12) is the same as the der ivativ e of the squareplus sho wn in Fig. 3, but shifted v ertically and scaled. 3) V ect or Activations: V ector activations output a vector whose components depend on all elements of the input, unlike elementwise activations, which appl y a scalar function 𝜏 m C 𝛙 s C g ( · ) 1 2 i s i ⊤ s J 𝛙 s g ( · ) (a) 𝛙 s 𝜏 m i s 𝜗 m cos ( 𝑘 𝜗 m ) sin ( 𝑘 𝜗 m ) i ⊤ s J 𝛙 s 𝑘 𝛝 ⊤ J 𝛕 𝛕 𝛝 g ( · ) ˜ x (b) Fig. 5. Proposed magnetic models in rotor coordinates: (a) without spatial harmonics, q-axis symmetry b y construction; (b) with spatial har monics. In both cases, the function g ( · ) is giv en by (10). independently to each input component. The scaled softmax activation is a common c hoice f or v ector activ ations, giv en by 𝛔 ( z ) = 1 Í 𝑁 𝑛 = 1 e 𝛽 𝑧 𝑛        e 𝛽 𝑧 1 . . . e 𝛽 𝑧 𝑁        (13) where the positive lear nable parameter 𝛽 affects the shape of the activation. Fig. 4(a) show s the shape of the first component 𝜎 1 ( 𝑧 1 , 𝑧 2 ) in the tw o-dimensional case. The softmax is the gradient of the log-sum-exp function 𝑆 ( z ) = log ( Í 𝑁 𝑛 = 1 e 𝛽 𝑧 𝑛 ) / 𝛽 . As sho wn in [26], the gradient netw ork (10) with the softmax activation can univ ersall y appro ximate any monotone conser v a- tiv e field. As a computationally more efficient alter nativ e to the softmax function, a 𝑝 -norm gradient can be used, given b y 𝛔 ( z ) = 1  1 + Í 𝑁 𝑛 = 1 ( 𝛽 𝑧 𝑛 ) 𝑝  𝑝 − 1 𝑝        ( 𝛽 𝑧 1 ) 𝑝 − 1 . . . ( 𝛽 𝑧 𝑁 ) 𝑝 − 1        (14) where 𝑝 is a positive ev en integer and 𝛽 is a positive lear nable parameter . This activation is the gradient of the smooth 𝑝 - norm 𝑆 ( z ) = [ 1 + Í 𝑁 𝑛 = 1 ( 𝛽 𝑧 𝑛 ) 𝑝 ] ( 1 / 𝑝 ) / 𝛽 , whic h is con v e x, thus guaranteeing monotonicity . The integer po wers can be calculated using simple multiplication, which is f ast. The onl y e xpensiv e operation is the fractional po w er , but it is needed only once per forward pass, regardless of the number 𝑁 of hidden units. In our context, the integer 𝑝 can typically be 6 or 8, cor responding to the e xponent values used in similar lo w- dimensional models f or iron saturation [3], [4], [20]. U nlike elementwise activations, the v ector activations (13) and (14) can be used with both cur rent and flux-linkage maps. The 𝑝 -nor m gradient offers computational efficiency and, in our e xperiments, pro vided comparable accuracy and robustness to the softmax. B. Incor por ating Physical Symmetries 1) Without Spatial Har monics: When spatial harmonics are omitted, the field energy depends onl y on the flux linkages. If 5 T ABLE I Ra ted V alues of the 5.6-kW PM Synchr onous Reluct ance Ma chine V oltage (line-to-neutral, peak v alue)  2 / 3 · 460 V 1.00 p.u. Current (peak value) √ 2 · 8 . 8 A 1.00 p.u. Frequency 60 Hz 1.00 p.u. Speed 1 800 r/min 1.00 p.u. Po w er 5.6 k W 0.80 p.u. T orque 29.7 Nm 0.80 p.u. the symmetry condition (6) is not enf orced, we may directly parametrize the cur rent map i s = g ( 𝛙 s ) using the monotone gradient netw ork (10). As discussed in Section II-B, the current map is monotone, corresponding to con v e xity of the field ener gy 𝑊 ( 𝛙 s ) . W e use A 0 = diag ( 𝜇 d , 𝜇 q ) to encode s trong conv e xity on the d- and q-axes. This architecture guarantees that the learned cur rent i s ( 𝛙 s ) is the g radient of a 𝜇 -strongly con v e x function, ensuring a unique current f or each flux linkage and enabling robust model in v ersion. The symmetr y condition (6) can be enf orced using the sym- metrized energy function (7). Differentiating (7) with respect to 𝛙 s giv es the cur rent map i s ( 𝛙 s ) = 1 2  g ( 𝛙 s ) + C g ( C 𝛙 s )  (15) This map i s ( 𝛙 s ) remains a monotone gradient netw ork, and its Jacobian is positive semidefinite. Fig. 5(a) sho ws the block diagram f or this magnetic model. By construction, the model satisfies q-axis symmetry . 2) With Spatial Harmonics: Spatial harmonics introduce angle dependence in the magnetic model (5), making both the current i s ( 𝛙 s , 𝜗 m ) and the torq ue term 𝜕 𝑊 / 𝜕 𝜗 m periodic in the rotor angle 𝜗 m . T o av oid discontinuities at the angle boundaries, w e use Fourier f eatures [27] 𝛝 =  cos ( 𝑘 𝜗 m ) sin ( 𝑘 𝜗 m )  (16a) where 𝑘 determines the electrical symmetry (e.g., 𝑘 = 6 f or a 60 ◦ electrical period). The field ener gy can then be expressed as 𝑊 ( 𝛙 s , 𝜗 m ) = ˜ 𝑊 ( 𝛙 s , 𝛝 ) . With this change of variables and the chain rule, the model (5) becomes i s =  𝜕 ˜ 𝑊 ( 𝛙 s , 𝛝 ) 𝜕 𝛙 s  ⊤ (16b) 𝜏 m = i ⊤ s J 𝛙 s + 𝑘 𝛝 ⊤ J  𝜕 ˜ 𝑊 ( 𝛙 s , 𝛝 ) 𝜕 𝛝  ⊤ (16c) Fig. 5(b) sho w s the block diag ram f or this magnetic model, where ˜ x = [ 𝛙 ⊤ s , 𝛝 ⊤ ] ⊤ is the combined input to the monotone gradient netw ork and 𝛕 = [ 𝜕 ˜ 𝑊 ( 𝛙 s , 𝛝 ) / 𝜕 𝛝 ] ⊤ is the gradient with respect to the Fourier f eatures. In the monotone g radient netw ork (10), w e use A 0 = diag ( 𝜇 d , 𝜇 q , 0 , 0 ) . This adds a linear ter m propor tional to the flux linkages to the output, which enf orces 𝜇 -strong monotonicity with respect to the flux linkages while leaving the angle f eatures unaffected. By construction, there e xis ts a scalar function ˜ 𝑊 ( ˜ x ) such that g ( ˜ x ) = [ 𝜕 ˜ 𝑊 ( ˜ x ) / 𝜕 ˜ x ] ⊤ , thus the gradient netw ork preser v es the conservativ e structure of the magnetic model. Fig. 6. T est bench including a 5.6-kW PM synchronous reluctance machine (left), load machine (right), and their inv erter cabinet (background). By representing the rotor angle through per iodic f eatures (16a), the field energy ˜ 𝑊 ( 𝛙 s , 𝛝 ) is modeled by a monotone gradient netw ork. As a result, the cur rent map i s ( 𝛙 s , 𝜗 m ) is strictly monotone in 𝛙 s f or each 𝜗 m , and both the current and torque are per iodic in 𝜗 m b y constr uction. This approach enf orces the monotonicity and per iodicity required by the ph ysical sy stem. IV . Resul ts Measured and FEM datasets from a four -pole 5.6-kW PM synchronous reluctance machine w ere used to validate the proposed magnetic models. T able I lists the rated v alues of the machine, and Fig. 6 show s the tes t bench used f or measurements. Due to the rotor flux bar r iers, a small effective air gap in the q- axis direction, and the presence of PMs, the machine has highl y nonlinear magnetic characteristics. The results are presented in per -unit v alues with base values derived from the rated v alues in T able I. T raining was implemented in Python using the PyT orc h library and AdamW optimizer . A. Measured Dataset Without Spatial Har monics 1) Dataset: The flux linkages of the ex ample machine were measured on an equidistant cur rent gr id using the constant-speed test [19]. By e xploiting q-axis symmetr y in the measurement procedure, the dataset contains 21 × 13 unique measurement points. Both d- and q-axis flux linkag es were measured at each point. Figs. 7 and 8 sho w this dataset in flux-linkage coordinates and cur rent coordinates, respectivel y . For illustration, both positiv e and negativ e q-axis values are shown, yielding a full gr id of 21 × 27 points in each map (567 display ed values per map). 2 2 The measured dataset is av ailable in an e xample of the motulator open-source project: https://aalto- electric- dr iv es.github.io/motulator/dr iv e e xamples/flux v ector/plot 6kw pmsyr m sat fvc.html. 6 0 . 0 0 . 5 1 . 0 𝜓 d (p.u.) − 1 . 5 0 . 0 1 . 5 𝜓 q (p.u.) − 2 0 2 4 𝑖 d (p.u.) (a) 0 . 0 0 . 5 1 . 0 𝜓 d (p.u.) − 1 . 5 0 . 0 1 . 5 𝜓 q (p.u.) − 4 − 2 0 2 4 𝑖 q (p.u.) (b) Fig. 7. Cur rent maps: (a) 𝑖 d ( 𝜓 d , 𝜓 q ) ; (b) 𝑖 q ( 𝜓 d , 𝜓 q ) . The sur f aces sho w the predicted maps from the 𝑝 -nor m gradient model (14) with 𝑝 = 8 and 𝑁 = 12 hidden units, trained on a 10% subset. Markers sho w the measured dataset: red indicates the 10% training subset and blue the remaining points. Gray lines sho w constant-current contours cor responding to the measured dataset. T ABLE II Comp arison of Activ ation Functions in Current Maps Without Spa tial Harmonics ( 𝑁 = 12 ) Activ ation T raining data 𝑒 rms (p.u.) 𝑒 max (p.u.) 𝑒 std (p.u.) squareplus (11) 10% 0.017 0.070 0.011 2% 0.076 0.344 0.054 softmax (13) 10% 0.031 0.226 0.021 2% 0.108 0.407 0.068 𝑝 -norm 10% 0.021 0.110 0.012 gradient (14) 2% 0.096 0.389 0.061 2) Model Configuration: Both energy-based current maps i s ( 𝛙 s ) and co-energy-based flux-linkage maps 𝛙 s ( i s ) w ere trained using the proposed approach. The q-axis symmetr y is enf orced b y construction, see Fig. 5(a). Different activation functions (11)–(14) are compared. The magnetic models without spatial har monics hav e two scalar in puts and tw o scalar outputs. The number of learnable parameters depends on the number 𝑁 of hidden units. The models ha v e an 𝑁 × 2 w eight matr ix A , an 𝑁 × 1 bias v ector b , a 2 × 2 diagonal output matrix A 0 , a 2 × 1 output bias b 0 , and a single learnable activation parameter 𝛽 . Consequentl y , the total number of learnable parameters is 3 𝑁 + 5. In the f ollo wing, we use 𝑁 = 12, resulting in 41 learnable parameters. This number of parameters is small compared to the lookup-table approach. − 2 − 1 0 1 2 𝑖 d (p.u.) − 2 − 1 0 1 2 𝑖 q (p.u.) 0 . 0 0 . 5 1 . 0 𝜓 d (p.u.) (a) − 2 − 1 0 1 2 𝑖 d (p.u.) − 2 − 1 0 1 2 𝑖 q (p.u.) − 1 . 5 0 . 0 1 . 5 𝜓 q (p.u.) (b) Fig. 8. Flux-linkage maps: (a) 𝜓 d ( 𝑖 d , 𝑖 q ) ; (b) 𝜓 q ( 𝑖 d , 𝑖 q ) . The surfaces show the predicted maps from the 𝑝 -norm gradient model (14) with 𝑝 = 8 and 𝑁 = 12 hidden units, trained on a 10% subset. Markers show the same measured dataset as in F ig. 7: red indicates the 10% training subset and blue the remaining points. Gray lines show constant-current contours cor responding to the measured dataset. T ABLE III Comp arison of Activ ation Functions in Flux -Linka ge Maps Without Sp a tial Harmonics ( 𝑁 = 12 ) Activ ation T raining data 𝑒 rms (p.u.) 𝑒 max (p.u.) 𝑒 std (p.u.) algebraic 10% 0.016 0.044 0.010 sigmoid (12) 2% 0.051 0.165 0.032 softmax (13) 10% 0.007 0.033 0.004 2% 0.029 0.081 0.019 𝑝 -norm 10% 0.004 0.022 0.003 gradient (14) 2% 0.018 0.061 0.012 The mean squared er ror (MSE) loss was used f or training current maps, defined as L = 1 𝐿 𝐿  ℓ = 1   i s ℓ − ˆ i s ℓ   2 (17) where 𝐿 is the number of training samples, i s ℓ is the measured current for sample ℓ , and ˆ i s ℓ ( 𝛙 s ℓ ) is the model prediction f or the corresponding flux-linkage input 𝛙 s ℓ . For training flux-linkag e maps, the same procedure was used, replacing the cur rent error with the flux-linkage error . 3) Model P er f ormance: Fig. 7 visualizes the lear ned cur rent maps with the 𝑝 -nor m gradient activ ation (14), when ev ery tenth data point (10% of the full measured dataset) is used for training and the rest f or validation. It can be obser v ed that the model v ery 7 accurately captures the measured data. Fig. 8 sho ws flux-linkage maps lear ned by training the co-energy -based dual model with the same activation and training dataset. The other activation functions yield visually similar results when trained with 10% of the dataset. For q uantitativ e compar ison, the root-mean-square (rms) error and the standard deviation o ver the entire measured dataset are computed, respectiv ely , as 𝑒 rms = v u t 1 𝐿 𝐿  ℓ = 1 𝑒 2 ℓ 𝑒 std = v u t 1 𝐿 𝐿  ℓ = 1 ( 𝑒 ℓ − ¯ 𝑒 ) 2 (18) where 𝑒 ℓ = ∥ i s ℓ − ˆ i s ℓ ∥ and ¯ 𝑒 = 1 𝐿 Í ℓ 𝑒 ℓ . Fur thermore, the maximum er ror 𝑒 max = max ℓ 𝑒 ℓ is considered. The abo v e metrics are defined for cur rent maps, but they can be similarly defined for flux-linkag e maps by replacing i s ℓ and ˆ i s ℓ with the corresponding flux linkages. T able II summar izes the results f or current maps when different activation functions are used. The table sho ws the results for tw o different training dataset sizes, 10% and 2% of the full measured dataset. In the latter case, ev ery 50th data point is used f or training, resulting in only 12 training samples per map. The results show that all models achie v e e xcellent accuracy when trained with 10% of the dataset and good results ev en with 2% of the dataset. These results indicate that the underl ying energy function can be well approximated with a sum of ridge functions. T able III gives the results for flux-linkag e maps. The models with v ector activations show better accuracy than those with elementwise activations, especially when the training dataset is limited to 2%. These results sugg est that the underl ying co- energy function may not be well approximated by a sum of ridge functions, thus requiring v ector activations f or accurate modeling with limited data. B. FEM Dataset With Spatial Har monics 1) Dataset: T o demonstrate the capability of the proposed model in Fig. 5(b) in captur ing spatial har monics, the same machine was analyzed using FEM under magnetostatic condi- tions on an equidistant g rid in cur rent and rotor angle. For this machine, the flux linkages and the torque are per iodic in the electrical angle 𝜗 m with a 60 ◦ period. The FEM dataset was generated on an equidistant cur rent gr id (61 × 61), ranging from − 2 . 41 to 2 . 41 p.u., and at 30 equidistant rotor angles from 0 ◦ to 60 ◦ . The total dataset size is 61 × 61 × 30 = 111 630 samples. At eac h operating point, the d- and q-axis flux linkages and the electromagnetic torque w ere computed. 2) Model Configuration: Both the energy-based and co- energy -based models w ere trained using the proposed approach. For brevity , the results are sho wn only f or the co-energy-based model, but the energy -based model show s similar per f ormance. The v ector activations softmax (13) and the 𝑝 -norm gradient (14) are compared. The training dataset size is either 10% or 0.2% of the full FEM dataset, corresponding to 11 163 and 223 samples, respectiv ely . The magnetic models with spatial har monics hav e f our scalar inputs and f our scalar outputs. In the f ollo wing e xamples, we use 0 1 2 𝑖 q (p.u.) 0 30 60 𝜗 m (deg) 0 . 0 0 . 5 1 . 0 1 . 5 𝜏 m (p.u.) (a) 0 15 30 45 60 𝜗 m (deg) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 𝜏 m (p.u.) Model V alidation T raining (b) Fig. 9. Electromagnetic torque from the softmax model with 𝑁 = 48 hidden units: (a) 𝜏 m ( 𝑖 q , 𝜗 m ) at constant 𝑖 d corresponding to the rated MTP A current; (b) 𝜏 m ( 𝜗 m ) at the constant rated MTP A stator current. The curve and markers in (b) are a slice from (a). Red and blue markers sho w the full measured dataset. Red mark ers indicate the 10% training subset. 𝑁 = 48 hidden units. Theref ore, the total number of learnable parameters is 5 𝑁 + 7 = 247. The MSE loss is used f or training, but it includes both the flux-linkage and torque errors. In the case of the co-energy-based models, the loss function is defined as L = 1 𝐿 𝐿  ℓ = 1 "   𝛙 s ℓ − ˆ 𝛙 s ℓ   2 𝜓 2 max +  𝜏 m ℓ − ˆ 𝜏 m ℓ  2 𝜏 2 max # (19) where 𝜓 max and 𝜏 max are the maximum flux linkage and torque in the training dataset, respectiv ely . When training energy -based models, the loss function is defined similar l y , replacing the flux linkage error term with the cur rent er ror term. 3) Model P er formance: Fig. 9(a) visualizes the proposed co- energy -based model with the softmax activation (13), trained with 10% of the FEM dataset. The model captures both the rotor -angle dependence and the q-axis cur rent dependence of the torque. Fig. 9(b) show s a slice at 𝑖 d = − 0 . 72 p.u. and 𝑖 q = 0 . 72 p.u., which approximatel y cor responds to the rated maximum-torque-per -ampere (MTP A) cur rent. The torque ripple caused by spatial harmonics is well captured. T able IV summar izes the prediction er rors f or flux linkage and torque ov er the entire FEM dataset. The model achiev es good accuracy e v en when the training dataset is limited to 0.2% of the full FEM dataset. The 𝑝 -norm gradient activation (14) giv es results comparable to the softmax activation (13) when 8 T ABLE IV Comp arison of Activ ation Functions with Sp a tial Harmonics ( 𝑁 = 48 ) Activ ation T raining Flux linkage error (p.u.) T orque er ror (p.u.) data 𝑒 rms 𝑒 max 𝑒 std 𝑒 rms 𝑒 max 𝑒 std softmax (13) 10% 0.008 0.035 0.005 0.012 0.077 0.008 0.2% 0.011 0.052 0.007 0.016 0.100 0.011 𝑝 -norm 10% 0.011 0.042 0.006 0.017 0.086 0.010 gradient (14) 0.2% 0.013 0.081 0.008 0.023 0.214 0.016 0 . 0 0 . 5 1 . 0 Speed (p.u.) 𝜔 ref m 𝜔 m 0 1 2 T orque (p.u.) 𝜏 ref m 𝜏 m ˆ 𝜏 m − 1 . 5 0 . 0 1 . 5 Current (p.u.) 𝑖 d 𝑖 q 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 0 . 30 Time (s) 0 . 0 0 . 5 1 . 0 Flux linkage (p.u.) 𝜓 ref s 𝜓 s ˆ 𝜓 s Fig. 10. Simulation e xample sho wing acceleration from s tandstill to 1 p.u. The machine model uses the dynamics (4) and the energy-based model (16) with spatial harmonics. Flux-vector control is applied, parametrized using the co- energy -based dual of (15). trained with 10% of the dataset, but yields slightly higher er rors when trained with 0.2% of the dataset. 4) Application Examples: The proposed models can be used f or various applications, suc h as control design and digital twins. Fig. 10 sho w s a simulation e x ample where the proposed energy - based magnetic model (16) with spatial harmonics is used. This f ormulation is con v enient in simulations, since the flux linkag es are states in the dynamics (4), resulting in a straightf orward implementation. In the simulation, a flux-v ector control sy stem [29], [30] uses the proposed co-ener gy-based dual model (i.e., the flux-linkag e maps) without spatial har monics. The flux-linkage maps are con v enient in control applications, since the measured stator current is a vailable while the flux linkages must be estimated. The optimal control loci are also computed from these flux- linkage maps. Fig. 11 show s the MTP A trajectory , the maximum- torque-per -v oltag e (MTPV) trajectory , and three cur rent limits. Owing to the smoothness of the model, these loci are smooth and free from numer ical noise, unlike in the case of linear interpolation models. The machine is accelerated from standstill to 1 p.u. speed, while the maximum cur rent is set to 2 p.u. The effect of spatial 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 𝜏 m (p.u.) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 𝜓 s (p.u.) MTP A MTPV 𝑖 s = 1 p.u. 𝑖 s = 2 p.u. 𝑖 s = 3 p.u. Fig. 11. Optimal control loci computed from the proposed model without spatial har monics, including the MTP A trajectory , the maximum-torque-per- v oltage (MTPV) tra jectory , and three current limits. harmonics is most apparent in the actual torque, but it is also visible in other signals. The proposed models allo w phy sicall y consistent high-fidelity simulations and provide a universal representation of the machine f or control design. V . Conclusions W e presented a phy sics-inf ormed magnetic modeling frame- w ork f or synchronous machines based on g radient netw orks. The stator cur rent and the electromagnetic torque are obtained as gradients of a single scalar field energy . This guarantees a conser v ativ e field with a symmetr ic Jacobian (reciprocity) b y construction, while the rotor angle input enables periodic dependence and spatial harmonics without breaking the gradient structure. Monotone gradient netw orks are used to enf orce single-valued mapping between the cur rent and flux linkage, corresponding to ph y sical properties of conv e x field energy and enabling robust inv ersion. If desired, q-axis symmetr y can also be enf orced. These ph y sical properties are embedded in the arc hitecture, impro ving data efficiency and pro viding smooth, ph ysicall y consistent predictions suitable for control and optimization applications, in addition to digital twins. Experiments on mea- sured and FEM data demonstrate that the proposed models are accurate and data-efficient. The presented modeling approach can be e xtended to other electric machine types. Appendix A Transforma tion to R otor Coordin a tes The same field energy can be e xpressed in different coor - dinates, 𝑊 ( 𝛙 s , 𝜗 m ) = 𝑊 s ( 𝛙 s s , 𝜗 m ) . Using the chain rule with 9 𝛙 s s = e 𝜗 m J 𝛙 s giv es 𝜕𝑊 ( 𝛙 s , 𝜗 m ) 𝜕 𝛙 s = 𝜕𝑊 s ( 𝛙 s s , 𝜗 m ) 𝜕 𝛙 s s 𝜕 𝛙 s s 𝜕 𝛙 s = ( i s s ) ⊤ e 𝜗 m J = i ⊤ s (20a) 𝜕𝑊 ( 𝛙 s , 𝜗 m ) 𝜕 𝜗 m = 𝜕𝑊 s ( 𝛙 s s , 𝜗 m ) 𝜕 𝛙 s s 𝜕 𝛙 s s 𝜕 𝜗 m + 𝜕𝑊 s ( 𝛙 s s , 𝜗 m ) 𝜕 𝜗 m = ( i s s ) ⊤ J 𝛙 s s + 𝜕𝑊 s ( 𝛙 s s , 𝜗 m ) 𝜕 𝜗 m (20b) Using 𝜏 m = − 𝜕𝑊 s / 𝜕 𝜗 m yields the e xpressions in (5). Appendix B J a cobian of the Gradient Network Model The v ector field (10) is conservativ e if its Jacobian J g is symmetric [26]. Applying the chain r ule, the Jacobian of the gradient network (10) becomes J g ( x ) = 𝜕 g ( x ) 𝜕 x = A 0 + A ⊤ 𝜕 𝛔 ( z ) 𝜕 z A (21) The first term A 0 is chosen to be symmetr ic. The second term is symmetric if J 𝛔 = 𝜕 𝛔 / 𝜕 z is symmetr ic, which is the case when the activation 𝛔 is a gradient of a scalar function. Notice that J 𝛔 is diagonal and thus symmetr ic when the activation is elementwise. The netw ork (10) is monotone if J g is positiv e semidefinite. This is guaranteed if A 0 and J 𝛔 are positive semidefinite. A ckno wledgments The authors thank Dr . Francesco Lelli, Ari Haa visto, and Hannu Har tikainen f or their contr ibutions regarding the mea- surements on the e xample machine, located at the EPE infras- tructure of Aalto School of Electrical Engineer ing. References [1] H. H. W oodson and J. R. Melcher , Electromechanical Dynamics . John Wile y & Sons, 1968. [Online]. A v ailable: https:// ocw .mit.edu/ ans7870/ resources/woodson/te xtbook/emd par t1.pdf [2] A. E. 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Ar mando, “Direct flux vector control of synchronous motor drives: A ccurate decoupled control with online adaptive maximum torque per ampere and maximum torque per v olts ev aluation, ” IEEE T rans. Ind. Electron. , vol. 69, no. 2, pp. 1235– 1243, Feb. 2022. 10 Junyi Li receiv ed the B.Sc. degree in communication engineering from the T aiyuan Univ ersity of T echnol- ogy , T aiyuan, China, in 2022, and the M.Sc. (T ech.) degree in information and communication engineer - ing from the Aalto Univ ersity , Espoo, Finland, in 2025. He is currently w orking tow ard the doctoral degree in automation and electr ical engineering at Aalto Univ ersity . His research interests include condition monitoring and phy sics-informed machine learning. Tim F oißner received the B.Sc. degrees in elec- trical engineering and inf ormation technology and in mechatronics from TU Dar mstadt, Dar mstadt, Germany , in 2023 and 2024, respectively . Since 2024, he has been pursuing the M.Sc. degree in automation and electr ical engineering with a specialization in electrical pow er engineer ing at Aalto Univ ersity , Espoo, Finland. Floran Martin receiv ed the Engineering Diploma in electrical engineering from Pol ytech Nantes, Nantes, France, in 2009, and the M.S. and Ph.D. degrees in electrical engineering from the Univ ersity of Nantes, Nantes, in 2009 and 2013, respectiv el y . In 2014, he joined the Depar tment of Electr ical En- gineering and Automation, Aalto Univ ersity , Espoo, Finland, where he is cur rentl y a Staff Scientist. His research interests include modeling of magnetic ma- terials as well as analyzing, designing, and controlling electrical machines. Antti Piippo received the M.Sc. (Eng.) and D.Sc. (T ech.) degrees in electrical engineer ing from the Helsinki Univ ersity of T echnology , Espoo, Finland, in 2003 and 2008, respectiv ely . He is cur rentl y an R&D Executiv e Engineer with ABB Oy , Drives, Helsinki, Finland. His main research interests include the control of electric drives. Mark o Hinkkanen (M’06–SM’13–F’23) received the M.Sc. (Eng.) and D.Sc. (T ech.) degrees in electrical engineer ing from the Helsinki U niv ersity of T ec hnology , Espoo, Finland, in 2000 and 2004, respectiv ely . He is cur rently a Full Professor with the School of Electrical Engineer ing, Aalto U niv ersity , Espoo, F in- land. His research interests include control systems, phy sics-inf ormed machine learning, electric machine drives, and po w er conv erters. Dr . Hinkkanen was the recipient of eight paper aw ards, including the 2016 Inter national Conf erence on Electrical Mac hines (ICEM) Brian J. Chalmers Best Paper A w ard, and the 2016 and 2018 IEEE Industry Applications Society Industrial Driv es Committee Bes t P aper A wards. He w as the corecipient of the 2020 SEMIKR ON Innov ation A ward. He w as the General Cochair of the 2018 IEEE 9th International Symposium on Sensorless Control for Electrical Drives (SLED). He is an Associate Editor of IEEE Transa ctions on Po wer Electronics .